{"id":6036,"date":"2023-04-05T16:58:34","date_gmt":"2023-04-05T07:58:34","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6036"},"modified":"2024-03-15T14:48:41","modified_gmt":"2024-03-15T05:48:41","slug":"sympy-%e3%81%a7-1-%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/sympy-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/sympy-%e3%81%a7-1-%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"SymPy \u3067 1 \u5909\u6570\u95a2\u6570\u306e\u5fae\u5206"},"content":{"rendered":"

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Python \u3067\u5fae\u5206\u7a4d\u5206\u306a\u3069\u306e\u8a18\u53f7\u8a08\u7b97\u3092\u3059\u308b\u5834\u5408\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092 import \u3057\u307e\u3059\u3002\uff08\u5f18\u5927 JupyterHub \u3067\u306f\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u306f\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u6e08\u307f\u3067\u3059\u3002\uff09<\/p>\n

\u30bb\u30eb\u3092\u30af\u30ea\u30c3\u30af\u3057\u3066\uff0c\u4e0a\u306e\u30e1\u30cb\u30e5\u30fc\u304b\u3089\u300c\u25b6\ufe0e Run\u300d\u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u304b\uff0cShift \u30ad\u30fc\u3092\u62bc\u3057\u306a\u304c\u3089 Enter \u30ad\u30fc\uff08Return \u30ad\u30fc\uff09\u3092\u62bc\u3059\u3068\u5b9f\u884c\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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from sympy.abc<\/span> import *<\/span> \r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9<\/h3>\n
$f(x) = x^2$ \u3092\u5fae\u5206\u3059\u308b\u3002<\/p>\n
Python \u3067\u306f\u95a2\u6570\u306e\u5b9a\u7fa9\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# Python \u3067\u306e\u95a2\u6570 f(x) \u306e\u5b9a\u7fa9<\/span>\r\n\r\ndef<\/span> f<\/span>(<\/span>x<\/span>):<\/span>\r\n    return<\/span> x<\/span>**<\/span>2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u307e\u305a\u306f\u4ee5\u4e0b\u306e\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u306b\u57fa\u3065\u3044\u3066\u6975\u9650\u3092\u8a08\u7b97\u3059\u308b\u3002
\n$$ \\frac{df}{dx} \\equiv \\lim_{h \\rightarrow 0} \\frac{f(x+h) – f(x)}{h}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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limit<\/span>((<\/span>f<\/span>(<\/span>x<\/span>+<\/span>h<\/span>)<\/span>-<\/span>f<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>h<\/span>,<\/span> h<\/span>,<\/span> 0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
$\\displaystyle 2 x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u3068\u3082\u304b\u304f\u7b54\u3048\u3092\u6c42\u3081\u305f\u3044\u5834\u5408\u306f\u3053\u308c\u3067 OK \u3067\u3059\u3002\u6975\u9650\u306e\u8a08\u7b97\u3092\u3059\u308b\u524d\u306e\u8868\u793a\u3092\u3057\u305f\u3044\u5834\u5408\u306f Limit()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002\u8a08\u7b97\u3092\u5b9f\u884c\u3055\u305b\u308b\u306b\u306f .doit()<\/code> \u3092\u3064\u3051\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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Eq<\/span>(<\/span>Limit<\/span>((<\/span>f<\/span>(<\/span>x<\/span>+<\/span>h<\/span>)<\/span>-<\/span>f<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>h<\/span>,<\/span> h<\/span>,<\/span> 0<\/span>),<\/span> \r\n   Limit<\/span>((<\/span>f<\/span>(<\/span>x<\/span>+<\/span>h<\/span>)<\/span>-<\/span>f<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>h<\/span>,<\/span> h<\/span>,<\/span> 0<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
$\\displaystyle \\lim_{h \\to 0^+}\\left(\\frac{- x^{2} + \\left(h + x\\right)^{2}}{h}\\right) = 2 x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u306e\u306b\u3044\u3061\u3044\u3061\u6975\u9650\u3092\u3068\u3063\u3066\u3044\u308b\u308f\u3051\u306b\u306f\u3044\u304d\u307e\u305b\u3093\u3002\u6b21\u306b\u5fae\u5206\u3092\u884c\u3046\u95a2\u6570 diff()<\/code> \u3092\u4f7f\u3063\u3066\u76f4\u63a5 f(x)<\/code> \u3092\u5fae\u5206\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
$\\displaystyle 2 x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5fae\u5206\u3092\u8a08\u7b97\u3059\u308b\u524d\u306e\u5f0f\u3092\u8868\u793a\u3055\u305b\u305f\u3044\u5834\u5408\u306f\uff0cDerivative()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
$\\displaystyle \\frac{d}{d x} x^{2} = 2 x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5fae\u5206\u6cd5\u306e\u516c\u5f0f<\/h3>\n
\u5fae\u5206\u6cd5\u306e\u516c\u5f0f<\/a> \u306b\u3042\u308b\u516c\u5f0f\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 1.\\ \\left\\{ f(x) \\pm g(x) \\right\\}’ = f'(x) \\pm g'(x)$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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# from sympy.abc import * \u3057\u3066\u3044\u308c\u3070\u4e0d\u8981\u3067\u3059\u304c<\/span>\r\n# \u5ff5\u306e\u305f\u3081<\/span>\r\nx<\/span> =<\/span> Symbol<\/span>(<\/span>'x'<\/span>)<\/span>\r\n\r\n# \u300c\u672a\u5b9a\u7fa9\u300d\u306a\u95a2\u6570\u306e\u5b9a\u7fa9<\/span>\r\nf<\/span> =<\/span> Function<\/span>(<\/span>'f'<\/span>)<\/span>\r\ng<\/span> =<\/span> Function<\/span>(<\/span>'g'<\/span>)<\/span>\r\n\r\n# \u3053\u308c\u3089\u306e\u95a2\u6570\u3092\u4f7f\u3046\u3068\u304d\u306e\u547c\u3073\u51fa\u3057\u65b9<\/span>\r\nf<\/span>(<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
$\\displaystyle f{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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# \u300c\u672a\u5b9a\u7fa9\u300d\u95a2\u6570\u306f\u3053\u306e\u3088\u3046\u306b\u3082\u5b9a\u7fa9\u3067\u304d\u308b<\/span>\r\nF<\/span> =<\/span> Function<\/span>(<\/span>'F'<\/span>)(<\/span>x<\/span>)<\/span>\r\n\r\n# \u3053\u308c\u3092\u547c\u3073\u51fa\u3059\u3068\u304d\u306b\u306f...<\/span>\r\nF<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
$\\displaystyle F{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span> +<\/span> g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span> +<\/span> g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\left(f{\\left(x \\right)} + g{\\left(x \\right)}\\right) = \\frac{d}{d x} f{\\left(x \\right)} + \\frac{d}{d x} g{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 2. \\ \\left\\{ c f(x) \\right\\}’ = c f'(x) $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>c<\/span>*<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>c<\/span>*<\/span>f<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} c f{\\left(x \\right)} = c \\frac{d}{d x} f{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u306e\u4f8b\u3067\u306f\u5de6\u8fba\u306b $\\displaystyle \\frac{\\partial}{\\partial x}$ \u3068\u3044\u3046\u5fae\u5206\u8a18\u53f7\u304c\u898b\u3048\u3066\u3044\u307e\u3059\u304c\uff0c\u3053\u308c\u306f\u504f\u5fae\u5206\u8a18\u53f7\u3067\u3059\u3002\u5225\u306e\u6388\u696d\u3067\u7fd2\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 3. \\ \\left\\{ f(x)\\, g(x) \\right\\}’ = f'(x)\\,g(x) + f(x)\\,g'(x) $<\/p>\n
\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>*<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>*<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
$\\displaystyle \\frac{d}{d x} f{\\left(x \\right)} g{\\left(x \\right)} = f{\\left(x \\right)} \\frac{d}{d x} g{\\left(x \\right)} + g{\\left(x \\right)} \\frac{d}{d x} f{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 4. \\ \\left\\{ \\frac{1}{g(x)} \\right\\}’ = – \\frac{g'(x)}{\\left\\{g(x)\\right\\}^2} $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>1<\/span>\/<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>1<\/span>\/<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\frac{1}{g{\\left(x \\right)}} = – \\frac{\\frac{d}{d x} g{\\left(x \\right)}}{g^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 5. \\ \\left\\{ \\frac{f(x)}{g(x)} \\right\\}’ = \\frac{f'(x)\\,g(x) – f(x)\\, g'(x)}{\\left\\{g(x)\\right\\}^2} $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\frac{f{\\left(x \\right)}}{g{\\left(x \\right)}} = – \\frac{f{\\left(x \\right)} \\frac{d}{d x} g{\\left(x \\right)}}{g^{2}{\\left(x \\right)}} + \\frac{\\frac{d}{d x} f{\\left(x \\right)}}{g{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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simplify()<\/code> \u306f\u901a\u5206\u306a\u3069\u3092\u884c\u306a\u3063\u3066\u300c\u7c21\u5358\u5316\u300d\u3057\u3066\u304f\u308c\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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simplify<\/span>(<\/span>diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>g<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
$\\displaystyle \\frac{- f{\\left(x \\right)} \\frac{d}{d x} g{\\left(x \\right)} + g{\\left(x \\right)} \\frac{d}{d x} f{\\left(x \\right)}}{g^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5408\u6210\u95a2\u6570\u306e\u5fae\u5206\u6cd5<\/h3>\n
$y = y(u), \\ u = u(x)$\uff0c\u3064\u307e\u308a $y$ \u304c $u$ \u3092\u901a\u3057\u3066 $x$ \u306e\u95a2\u6570\u3067\u3042\u308b\u3068\u304d\uff0c<\/p>\n
$$\\frac{dy}{dx} = \\frac{dy}{du} \\frac{du}{dx}$$<\/p>\n
\u3068\u3044\u3046\u3053\u3068\u3092\u7406\u89e3\u3057\u3066\u3044\u308b\u304b\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\nu<\/span> =<\/span> Function<\/span>(<\/span>'u'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>y(u(x))<\/span>,<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>y(u(x))<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(u{\\left(x \\right)} \\right)} = \\frac{d}{d x} u{\\left(x \\right)} \\frac{d}{d u{\\left(x \\right)}} y{\\left(u{\\left(x \\right)} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\uff1a $ y = (a x^2 + b x + c)^5 $ \u306e\u5fae\u5206\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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u<\/span> =<\/span> a<\/span>*<\/span>x<\/span>**<\/span>2<\/span> +<\/span> b<\/span>*<\/span>x<\/span> +<\/span> c<\/span>\r\ny<\/span> =<\/span> u<\/span>**<\/span>5<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} \\left(a x^{2} + b x + c\\right)^{5} = \\left(10 a x + 5 b\\right) \\left(a x^{2} + b x + c\\right)^{4}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\n
\u3079\u304d\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n
\u3079\u304d\u4e57\uff08\u7d2f\u4e57\uff09\u306f **<\/code> \u3067\u8868\u3057\u307e\u3059\u3002$x^p$ \u306f\uff0cx**p<\/code>\u3002\u3079\u304d\u95a2\u6570 x**p<\/code> \u3092\u5fae\u5206\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
_<\/code> \u306f\u76f4\u524d\u306e\u8a08\u7b97\u7d50\u679c\u3092\u3042\u3089\u308f\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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diff<\/span>(<\/span>x<\/span>**<\/span>p<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
$\\displaystyle \\frac{p x^{p}}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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simplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[20]:<\/div>\n
$\\displaystyle p x^{p – 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\n
\u4f8b\u3068\u3057\u3066\uff0c\u3079\u304d\u95a2\u6570\u306e\u5408\u6210\u95a2\u6570\u3067\u3042\u308b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u95a2\u6570\u306e\u5fae\u5206\u3002
\n$$\\frac{d}{dx} \\frac{1}{\\sqrt{1 + x^2}}$$<\/p>\n
$\\sqrt{x} =$ sqrt(x)<\/code> \u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[21]:<\/div>\n
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f<\/span> =<\/span> 1<\/span>\/<\/span>sqrt<\/span>(<\/span>1<\/span> +<\/span> x<\/span>**<\/span>2<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>f<\/span>,<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
$\\displaystyle \\frac{d}{d x} \\frac{1}{\\sqrt{x^{2} + 1}} = – \\frac{x}{\\left(x^{2} + 1\\right)^{\\frac{3}{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u53c2\u8003\uff1a\u3079\u304d\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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$y = x^2$ \u3092 $ -3 \\leq x 3$ \u306e\u7bc4\u56f2\u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u4f8b\u30022\u6b21\u5143\u30b0\u30e9\u30d5\u306f plot()<\/code> \u95a2\u6570\u3067\u63cf\u304d\u307e\u3059\u3002\u5f18\u5927 JupyterHub \u3067\u306f SymPy Plotting Backends<\/a> \u3082\u30a4\u30f3\u30b9\u30c8\u30fc\u30eb\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b import \u3057\u3066\u3044\u308b\u3053\u3068\u3092\u524d\u63d0\u306b\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n
\n
# SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/div>\n
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plot<\/span>(<\/span>x<\/span>**<\/span>2<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>,<\/span> 3<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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<\/p>\n
\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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$y = x, x^2, x^3$ \u3092 $ -3 \\leq x 3$ \u306e\u7bc4\u56f2\u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u4f8b\u3002<\/p>\n
    \n
  • $y$ \u8ef8\uff08\u7e26\u8ef8\uff09\u306e\u8868\u793a\u7bc4\u56f2\u3092\u8a2d\u5b9a\u3059\u308b\u3068\u304d\u306f ylim = (ymin, ymax)<\/code> \u306a\u3069\u306e\u3088\u3046\u306b\u3057\u307e\u3059\u3002<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[23]:<\/div>\n
    \n
    \n
    \n
    plot<\/span>(<\/span>x<\/span>,<\/span> x<\/span>**<\/span>2<\/span>,<\/span> x<\/span>**<\/span>3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> \r\n     ylim<\/span> =<\/span> (<\/span>-<\/span>3<\/span>,<\/span> 7<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    $y = x, x^{1\/2}, x^{1\/3}$ \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u4f8b\u3067\u3059\u3002<\/p>\n
    \uff08\u7d30\u304b\u306a\u3053\u3068\u3067\u3059\u304c\uff0cPython \u3067\u306f 1\/3 = 0.3333333333333333<\/code> \u306a\u306e\u3067\u5206\u6570\u306e\u307e\u307e\u306b\u3059\u308b\u305f\u3081\u306b Rational(1, 3)<\/code> \u306a\u3069\u3068\u3057\u3066\u3044\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[24]:<\/div>\n
    \n
    \n
    \n
    plot<\/span>(<\/span>x<\/span>,<\/span> x<\/span>**<\/span>(<\/span>Rational<\/span>(<\/span>1<\/span>,<\/span> 2<\/span>)),<\/span> x<\/span>**<\/span>(<\/span>Rational<\/span>(<\/span>1<\/span>,<\/span>3<\/span>)),<\/span> (<\/span>x<\/span>,<\/span> 0<\/span>,<\/span> 3<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    $y = x^{-2}, \\ x^{-3}$ \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u4f8b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[25]:<\/div>\n
    \n
    \n
    \n
    plot<\/span>(<\/span>1<\/span>\/<\/span>x<\/span>**<\/span>2<\/span>,<\/span> 1<\/span>\/<\/span>x<\/span>**<\/span>3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    $y = x^{-3}$ \u306e\u30b0\u30e9\u30d5\u304c\u5909\u3067\u3059\u3002$y \\rightarrow -\\infty$ \u3068 $y \\rightarrow +\\infty$ \u3092\u3064\u306a\u3050\u7e26\u306e\u7dda\u304c\u898b\u3048\u3066\u3044\u307e\u3059\u3002\u4e0d\u9023\u7d9a\u70b9\u306e\u51e6\u7406\u3092\u9069\u5207\u306b\u884c\u3046\u305f\u3081\u306b\uff0cdetect_poles = True<\/code> \u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u3064\u3051\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[26]:<\/div>\n
    \n
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    plot<\/span>(<\/span>1<\/span>\/<\/span>x<\/span>**<\/span>2<\/span>,<\/span> 1<\/span>\/<\/span>x<\/span>**<\/span>3<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \r\n     detect_poles<\/span> =<\/span> True<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u6307\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n
    \u5ff5\u306e\u305f\u3081\uff0cSymPy \u304c $\\displaystyle \\lim_{h \\rightarrow 0}(1 + h)^{\\frac{1}{h}}$ \u304c\u308f\u304b\u308b\u304b\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    In\u00a0[27]:<\/div>\n
    \n
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    Eq<\/span>(<\/span>Limit<\/span>((<\/span>1<\/span> +<\/span> h<\/span>)<\/span>**<\/span>(<\/span>1<\/span>\/<\/span>h<\/span>),<\/span> h<\/span>,<\/span> 0<\/span>),<\/span> \r\n   Limit<\/span>((<\/span>1<\/span> +<\/span> h<\/span>)<\/span>**<\/span>(<\/span>1<\/span>\/<\/span>h<\/span>),<\/span> h<\/span>,<\/span> 0<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[27]:<\/div>\n
    $\\displaystyle \\lim_{h \\to 0^+} \\left(h + 1\\right)^{\\frac{1}{h}} = e$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u30cd\u30a4\u30d4\u30a2\u6570 $e$ \u3092\u5e95\u3068\u3057\u305f\u6307\u6570\u95a2\u6570 $e^x$ \u306f exp(x)<\/code> \u3067\u8868\u3057\u307e\u3059\u3002<\/p>\n
    $$\\frac{d}{dx} e^x = \\cdots$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[28]:<\/div>\n
    \n
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    Eq<\/span>(<\/span>Derivative<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[28]:<\/div>\n
    $\\displaystyle \\frac{d}{d x} e^{x} = e^{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u5fae\u5206\u3057\u305f\u7b54\u3048\u3060\u3051\u3092\u77e5\u308a\u305f\u3044\u306a\u3089\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3059\u308b\u3060\u3051\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[29]:<\/div>\n
    \n
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    diff<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
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    Out[29]:<\/div>\n
    $\\displaystyle e^{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u53c2\u8003\uff1aSymPy \u306e\u57fa\u672c\u7684\u306a\u5b9a\u6570\u3068\u3057\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n\n\n
    \u3000\u5186\u5468\u7387 $\\pi$\u3000<\/span><\/th>\n\u3000\u30cd\u30a4\u30d4\u30a2\u6570 $e$\u3000<\/span><\/th>\n\u3000\u865a\u6570\u5358\u4f4d $i$\u3000<\/span><\/th>\n\u3000\u7121\u9650\u5927 $\\infty$\u3000<\/span><\/th>\n<\/tr>\n<\/thead>\n
    pi<\/code><\/td>\nE<\/code><\/td>\nI<\/code><\/td>\noo<\/code><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    $\\exp(1) = e^1 =$ E<\/code> \u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    In\u00a0[30]:<\/div>\n
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    exp<\/span>(<\/span>1<\/span>)<\/span> -<\/span> E<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
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    Out[30]:<\/div>\n
    $\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    float()<\/code> \u95a2\u6570\u3067\u30cd\u30a4\u30d4\u30a2\u6570\u306e\u5024\u3092\u5c0f\u6570\u70b9\u8868\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[31]:<\/div>\n
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    float<\/span>(<\/span>E<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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    Out[31]:<\/div>\n
    \n
    2.718281828459045<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u53c2\u8003\uff1a\u6307\u6570\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n
    $y = e^x, e^{-x}$ \u306e\u30b0\u30e9\u30d5\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    In\u00a0[32]:<\/div>\n
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    plot<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>),<\/span> exp<\/span>(<\/span>-<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>,<\/span> 3<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
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    \n
    <\/div>\n
    \n
    <\/p>\n
    \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u5bfe\u6570\u95a2\u6570<\/h3>\n
    \u5b9a\u7fa9<\/h4>\n
    \u81ea\u7136\u5bfe\u6570\u306f log(x)<\/code> \u3067\u3059\u3002\u81ea\u7136\u5bfe\u6570\u306f\uff0c\u30cd\u30a4\u30d4\u30a2\u6570 $e$ \u3092\u5e95\u3068\u3057\u305f\u6307\u6570\u95a2\u6570 $e^x$ \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u304b\u3089\uff0c<\/p>\n
    $$e^{\\log x} = x$$<\/p>\n
    \u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[33]:<\/div>\n
    \n
    \n
    \n
    exp<\/span>(<\/span>log<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    \n
    Out[33]:<\/div>\n
    $\\displaystyle x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u5fae\u5206<\/h4>\n
    $$\\frac{d}{dx} \\log x = \\cdots$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
    In\u00a0[34]:<\/div>\n
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    \n
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    diff<\/span>(<\/span>log<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    \n
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    Out[34]:<\/div>\n
    $\\displaystyle \\frac{1}{x}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
    \n
    <\/div>\n
    \n
    \n
    \u53c2\u8003\uff1a\u5bfe\u6570\u6cd5\u5247<\/h4>\n
      \n
    1. $\\log(x y) = \\log x + \\log y$<\/li>\n
    2. $\\log(x^y) = y \\log x$<\/li>\n<\/ol>\n
      \u3053\u308c\u3089\u3092\u78ba\u8a8d\u3059\u308b\u306b\u306f expand_log()<\/code> \u95a2\u6570\u306b force=True<\/code> \u30aa\u30d7\u30b7\u30e7\u30f3\u304c\u5fc5\u8981\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[35]:<\/div>\n
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      # x, y = symbols('x y')<\/span>\r\nvar<\/span>(<\/span>'x y'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[35]:<\/div>\n
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      (x, y)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[36]:<\/div>\n
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      expand_log<\/span>(<\/span>log<\/span>(<\/span>x<\/span>*<\/span>y<\/span>),<\/span> force<\/span>=<\/span>True<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[36]:<\/div>\n
      $\\displaystyle \\log{\\left(x \\right)} + \\log{\\left(y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[37]:<\/div>\n
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      expand_log<\/span>(<\/span>log<\/span>(<\/span>x<\/span>**<\/span>y<\/span>),<\/span> force<\/span>=<\/span>True<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[37]:<\/div>\n
      $\\displaystyle y \\log{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u3055\u3089\u306b\uff0c$\\log x$ \u306f $e^x$ \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089<\/p>\n
      $$\\log e^x = x$$<\/p>\n
      \u3082\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u78ba\u8a8d\u3059\u308b\u306b\u306f\uff0cforce=True<\/code> \u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u3064\u3051\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[38]:<\/div>\n
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      expand_log<\/span>(<\/span>log<\/span>(<\/span>exp<\/span>(<\/span>x<\/span>)),<\/span> force<\/span>=<\/span>True<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[38]:<\/div>\n
      $\\displaystyle x$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u53c2\u8003\uff1a\u5bfe\u6570\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n
      $y = \\log x$ \u306e\u30b0\u30e9\u30d5\u3092 $0 < x < 5$ \u306e\u7bc4\u56f2\u3067\u63cf\u304f\u3002$y$ \u306e\u8868\u793a\u7bc4\u56f2\u3092 $ -3 < y < 3$ \u304f\u3089\u3044\u306b\u3057\u3066\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[39]:<\/div>\n
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      plot<\/span>(<\/span>log<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> 0.01<\/span>,<\/span> 5<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>3<\/span>,<\/span> 3<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u4e09\u89d2\u95a2\u6570<\/h3>\n
      \u6027\u8cea<\/h4>\n
      \u4e09\u89d2\u95a2\u6570\u306f sin(x), cos(x), tan(x)<\/code> \u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n
      $\\sin^2 x + \\cos^2 x = 1$ \u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[40]:<\/div>\n
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      simplify<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span> +<\/span> cos<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[40]:<\/div>\n
      $\\displaystyle 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u6975\u9650\u516c\u5f0f\u306e\u78ba\u8a8d<\/h4>\n
      \u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206\u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u6975\u9650<\/p>\n
      $$ \\lim_{x \\rightarrow 0} \\frac{\\sin x}{x} = \\cdots$$<\/p>\n
      \u306e\u5024\u304c\u5fc5\u8981\u3067\u3057\u305f\u3002\u3053\u306e\u6975\u9650\u304c\u3067\u304d\u308b\u304b\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[41]:<\/div>\n
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      Eq<\/span>(<\/span>Limit<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>,<\/span> 0<\/span>),<\/span> \r\n   limit<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>,<\/span> 0<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[41]:<\/div>\n
      $\\displaystyle \\lim_{x \\to 0^+}\\left(\\frac{\\sin{\\left(x \\right)}}{x}\\right) = 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      $\\displaystyle \\lim_{x \\rightarrow 0} \\frac{1 – \\cos x}{x} = 0$ \u3067\u3042\u308b\u3053\u3068\u3082\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[42]:<\/div>\n
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      Eq<\/span>(<\/span>Limit<\/span>((<\/span>1<\/span>-<\/span>cos<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>,<\/span> 0<\/span>),<\/span> \r\n   limit<\/span>((<\/span>1<\/span>-<\/span>cos<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>x<\/span>,<\/span> x<\/span>,<\/span> 0<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[42]:<\/div>\n
      $\\displaystyle \\lim_{x \\to 0^+}\\left(\\frac{1 – \\cos{\\left(x \\right)}}{x}\\right) = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u52a0\u6cd5\u5b9a\u7406\u306e\u78ba\u8a8d<\/h4>\n
      $\\sin (x + h) = \\sin x \\, \\cos h + \\cos x \\, \\sin h$ \u304a\u3088\u3073<\/p>\n
      $\\cos (x + h) = \\cos x \\, \\cos h – \\sin x \\, \\sin h$ \u306e\u78ba\u8a8d\u3002<\/p>\n
      \u4e09\u89d2\u95a2\u6570\u3092\u300c\u5c55\u958b\u300d\u3059\u308b\u3068\u304d\u306f\uff0cexpand()<\/code> \u95a2\u6570\u306b trig = True<\/code> \u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u3064\u3051\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[43]:<\/div>\n
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      expand<\/span>(<\/span>sin<\/span>(<\/span>x<\/span> +<\/span> h<\/span>),<\/span> trig<\/span> =<\/span> True<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[43]:<\/div>\n
      $\\displaystyle \\sin{\\left(h \\right)} \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)} \\cos{\\left(h \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[44]:<\/div>\n
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      expand<\/span>(<\/span>cos<\/span>(<\/span>x<\/span> +<\/span> h<\/span>),<\/span> trig<\/span> =<\/span> True<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[44]:<\/div>\n
      $\\displaystyle – \\sin{\\left(h \\right)} \\sin{\\left(x \\right)} + \\cos{\\left(h \\right)} \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
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      \u5fae\u5206<\/h4>\n
      \u307e\u305a\u306f\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u306b\u5f93\u3063\u3066 $\\sin x$ \u306e\u5fae\u5206\u3092\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
      $\\displaystyle \\frac{d}{dx} \\sin x = \\lim_{h \\rightarrow 0} \\frac{\\sin(x+h) – \\sin (x)}{h} = \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[45]:<\/div>\n
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      Eq<\/span>(<\/span>Limit<\/span>((<\/span>sin<\/span>(<\/span>x<\/span>+<\/span>h<\/span>)<\/span>-<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>h<\/span>,<\/span> h<\/span>,<\/span> 0<\/span>),<\/span> \r\n   limit<\/span>((<\/span>sin<\/span>(<\/span>x<\/span>+<\/span>h<\/span>)<\/span>-<\/span>sin<\/span>(<\/span>x<\/span>))<\/span>\/<\/span>h<\/span>,<\/span> h<\/span>,<\/span> 0<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[45]:<\/div>\n
      $\\displaystyle \\lim_{h \\to 0^+}\\left(\\frac{- \\sin{\\left(x \\right)} + \\sin{\\left(h + x \\right)}}{h}\\right) = \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      diff()<\/code> \u95a2\u6570\u3067\u4e09\u89d2\u95a2\u6570\u3092\u5fae\u5206\u3057\u3066\u3082\u540c\u3058\u7b54\u3048\u304c\u51fa\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[46]:<\/div>\n
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      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[46]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\sin{\\left(x \\right)} = \\cos{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      $\\cos x, \\tan x$ \u306e\u5fae\u5206\u306b\u3064\u3044\u3066\u3082 diff()<\/code> \u95a2\u6570\u3067\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[47]:<\/div>\n
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      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>cos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[47]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\cos{\\left(x \\right)} = – \\sin{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[48]:<\/div>\n
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      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>tan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>tan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[48]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\tan{\\left(x \\right)} = \\tan^{2}{\\left(x \\right)} + 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u53f3\u8fba\u306f\u3053\u308c\u3067\u3082\u3088\u3044\u3067\u3059\u304c\uff0csimplify()<\/code> \u95a2\u6570\u3067\u300c\u7c21\u5358\u5316\u300d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[49]:<\/div>\n
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      simplify<\/span>(<\/span>diff<\/span>(<\/span>tan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Out[49]:<\/div>\n
      $\\displaystyle \\frac{1}{\\cos^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u53c2\u8003\uff1a\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n
      $y = \\sin x, \\cos x, \\tan x$ \u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304d\u307e\u3059\u3002$ -2 \\pi \\leq x \\leq 2 \\pi$ \u306e\u7bc4\u56f2\u3067\u3002$\\tan x$ \u306f\u975e\u5e38\u306b\u5927\u304d\u3044\u5024\u306b\u306a\u308b\u5834\u5408\u304c\u3042\u308b\u306e\u3067\uff0c$y$ \u306e\u8868\u793a\u7bc4\u56f2\u3092\u9069\u5b9c\u8a2d\u5b9a\u3057\u307e\u3059\u3002<\/p>\n
      \u307e\u305f\uff0c$y = \\tan x $ \u306e\u4e0d\u9023\u7d9a\u70b9\u3092\u9069\u5207\u306b\u51e6\u7406\u3059\u308b\u305f\u3081\u306b detect_poles = True<\/code> \u30aa\u30d7\u30b7\u30e7\u30f3\u3092\u3064\u3051\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[50]:<\/div>\n
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      plot<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> cos<\/span>(<\/span>x<\/span>),<\/span> tan<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>2<\/span>*<\/span>pi<\/span>,<\/span> 2<\/span>*<\/span>pi<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> \r\n     detect_poles<\/span> =<\/span> True<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      <\/div>\n
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      \u53c2\u8003\uff1a\u6a2a\u8ef8\u306e\u76ee\u76db\u3092 $\\pi\/2$ \u3054\u3068\u306b<\/h4>\n
      $x$ \u8ef8\uff08\u6a2a\u8ef8\uff09\u306e\u76ee\u76db\u3092 $\\pi\/2$ \u3054\u3068\u306b\u3057\u3066\u307f\u308b\u3002\u307e\u305f SPB\/Matplotlib \u306e\u30c7\u30d5\u30a9\u30eb\u30c8\u3067\u306f\u30b0\u30ea\u30c3\u30c9\u7dda\u304c\u5b9f\u7dda\u3067\u3061\u3068\u76ee\u969c\u308a\u3060\u3063\u305f\u308a\uff0c$x$ \u8ef8\uff0c$y$ \u8ef8\u304c\u8868\u793a\u3055\u308c\u306a\u304b\u3063\u305f\u308a\u306a\u306e\u3067\uff0c\u305d\u306e\u8fba\u3082\u8a2d\u5b9a\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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      In\u00a0[51]:<\/div>\n
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      \n
      \n
      p<\/span> =<\/span> plot<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> cos<\/span>(<\/span>x<\/span>),<\/span> tan<\/span>(<\/span>x<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>2<\/span>*<\/span>pi<\/span>,<\/span> 2<\/span>*<\/span>pi<\/span>),<\/span>\r\n         xlim<\/span> =<\/span> (<\/span>-<\/span>2<\/span>*<\/span>pi<\/span>,<\/span> 2<\/span>*<\/span>pi<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> \r\n         detect_poles<\/span> =<\/span> True<\/span>,<\/span> size<\/span> =<\/span> (<\/span>6.28<\/span>,<\/span> 6.4<\/span>),<\/span>\r\n         show<\/span> =<\/span> False<\/span>)<\/span>\r\nax<\/span> =<\/span> p<\/span>.<\/span>ax<\/span>\r\n\r\n# x \u8ef8\u306e\u76ee\u76db\u306f 0.5\u03c0 \u3054\u3068\u306b<\/span>\r\nPi<\/span> =<\/span> float<\/span>(<\/span>pi<\/span>)<\/span>\r\nax<\/span>.<\/span>set_xticks<\/span>(<\/span>\r\n    [(<\/span>0.5<\/span>*<\/span>i<\/span>*<\/span>Pi<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>4<\/span>,<\/span> 5<\/span>)])<\/span>\r\n# y \u8ef8\u306e\u76ee\u76db\u306f 1 \u3054\u3068\u306b<\/span>\r\nax<\/span>.<\/span>set_yticks<\/span>(<\/span>\r\n    [<\/span>i<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>5<\/span>,<\/span> 6<\/span>)])<\/span>\r\n\r\n# x \u8ef8\u306e\u76ee\u76db\u306e\u30e9\u30d9\u30eb\u3002\u65e2\u7d04\u5206\u6570\u8868\u8a18\u3067\u3002<\/span>\r\nax<\/span>.<\/span>set_xticklabels<\/span>(<\/span>\r\n    [(<\/span>str<\/span>(<\/span>Rational<\/span>(<\/span>i<\/span>,<\/span>2<\/span>))<\/span>+<\/span>\" $\\pi$\"<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>4<\/span>,<\/span> 5<\/span>)])<\/span>\r\n\r\n# \u30b0\u30ea\u30c3\u30c9\u3092\u7070\u8272\u70b9\u7dda\u306b<\/span>\r\nax<\/span>.<\/span>grid<\/span>(<\/span>True<\/span>,<\/span> which<\/span>=<\/span>\"major\"<\/span>,<\/span> color<\/span>=<\/span>\"lightgray\"<\/span>,<\/span> \r\n        dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 5<\/span>),<\/span> linewidth<\/span>=<\/span>0.5<\/span>)<\/span>\r\n\r\n# x\u8ef8 y\u8ef8\u3002\u8868\u793a\u30aa\u30d7\u30b7\u30e7\u30f3\u306f\u306a\u3044\u306e\u3067\u3053\u308c\u3067\u4ee3\u7528\u3002<\/span>\r\nax<\/span>.<\/span>axhline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'black'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 5<\/span>),<\/span> linewidth<\/span>=<\/span>0.5<\/span>)<\/span>\r\nax<\/span>.<\/span>axvline<\/span>(<\/span>0<\/span>,<\/span> color<\/span>=<\/span>'black'<\/span>,<\/span> dashes<\/span>=<\/span>(<\/span>3<\/span>,<\/span> 5<\/span>),<\/span> linewidth<\/span>=<\/span>0.5<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u9006\u4e09\u89d2\u95a2\u6570<\/h3>\n
      \u5b9a\u7fa9<\/h4>\n
      \u9006\u4e09\u89d2\u95a2\u6570\u306f\u4e09\u89d2\u95a2\u6570\u306e\u9006\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\uff0c\u305d\u308c\u305e\u308c<\/p>\n
      $\\sin^{-1} x = \\arcsin x = $ asin(x)<\/code>
      \n$\\cos^{-1} x = \\arccos x = $ acos(x)<\/code>
      \n$\\tan^{-1} x = \\arctan x = $ atan(x)<\/code><\/p>\n
      \u3068\u66f8\u304d\u307e\u3059\u3002\u9006\u95a2\u6570\u3067\u3059\u304b\u3089\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u306f\u305a\u3067\u3059\u3002<\/p>\n
      $\\sin\\left(\\sin^{-1} x\\right) = x$
      \n$\\cos\\left(\\cos^{-1} x\\right) = x$
      \n$\\tan\\left(\\tan^{-1} x\\right) = x$<\/p>\n
      \u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[52]:<\/div>\n
      \n
      \n
      \n
      sin<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>)),<\/span> cos<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>)),<\/span> tan<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[52]:<\/div>\n
      \n
      (x, x, x)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u3057\u304b\u3057\uff0c$\\sin^{-1} \\left(\\sin x\\right) = x$ \u306a\u3069\u3068\u306f\u4e00\u822c\u306b\u306f\u306a\u308a\u307e\u305b\u3093\u3002<\/p>\n
      \u53cd\u4f8b\u306f $x = \\pi$ \u3068\u3059\u308c\u3070…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[53]:<\/div>\n
      \n
      \n
      \n
      asin<\/span>(<\/span>sin<\/span>(<\/span>pi<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[53]:<\/div>\n
      $\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u5fae\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[54]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>asin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[54]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\operatorname{asin}{\\left(x \\right)} = \\frac{1}{\\sqrt{1 – x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[55]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>acos<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[55]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\operatorname{acos}{\\left(x \\right)} = – \\frac{1}{\\sqrt{1 – x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[56]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>atan<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[56]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\operatorname{atan}{\\left(x \\right)} = \\frac{1}{x^{2} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u53c2\u8003\uff1a\u9006\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n
      $\\sin^{-1} x$ \u3068 $\\cos^{-1} x$ \u306e\u5b9a\u7fa9\u57df\u306f $ -1 < x < 1$\uff0c$\\tan^{-1} x$ \u306e\u5b9a\u7fa9\u6a5f\u306f $-\\infty < x < \\infty$ \u3067\u3042\u308b\u3053\u3068\u306b\u7559\u610f\u3057\u3066\uff0c\u9006\u4e09\u89d2\u95a2\u6570\u306e\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3002<\/p>\n
      $y$ \u8ef8\uff08\u7e26\u8ef8\uff09\u306e\u76ee\u76db\u3092 $\\pi\/4$ \u3054\u3068\u306b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[57]:<\/div>\n
      \n
      \n
      \n
      p<\/span> =<\/span> plot<\/span>((<\/span>asin<\/span>(<\/span>x<\/span>),<\/span> \"arcsin $x$\"<\/span>),<\/span> (<\/span>acos<\/span>(<\/span>x<\/span>),<\/span> \"arccos $x$\"<\/span>),<\/span> \r\n         (<\/span>x<\/span>,<\/span> -<\/span>1<\/span>,<\/span> 1<\/span>),<\/span> xlim<\/span> =<\/span> (<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>),<\/span> show<\/span> =<\/span> False<\/span>)<\/span>\r\n\r\nax<\/span> =<\/span> p<\/span>.<\/span>ax<\/span>\r\n\r\n# y \u8ef8\u306e\u76ee\u76db<\/span>\r\nPi<\/span> =<\/span> float<\/span>(<\/span>pi<\/span>)<\/span>\r\nax<\/span>.<\/span>set_yticks<\/span>(<\/span>\r\n    [(<\/span>0.25<\/span>*<\/span>i<\/span>*<\/span>Pi<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>2<\/span>,<\/span> 5<\/span>)])<\/span>\r\n# y \u8ef8\u306e\u30e9\u30d9\u30eb<\/span>\r\nax<\/span>.<\/span>set_yticklabels<\/span>(<\/span>\r\n    [<\/span>\"<\/span>%.2f<\/span> $\\pi$\"<\/span> %<\/span> (<\/span>0.25<\/span>*<\/span>i<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>2<\/span>,<\/span> 5<\/span>)]);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[58]:<\/div>\n
      \n
      \n
      \n
      p<\/span> =<\/span> plot<\/span>((<\/span>atan<\/span>(<\/span>x<\/span>),<\/span> \"arctan $x$\"<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>10<\/span>,<\/span> 10<\/span>),<\/span> legend<\/span> =<\/span> True<\/span>,<\/span>\r\n         xlim<\/span> =<\/span> (<\/span>-<\/span>10<\/span>,<\/span> 10<\/span>),<\/span> ylim<\/span> =<\/span> (<\/span>-<\/span>pi<\/span>\/<\/span>2<\/span>,<\/span> pi<\/span>\/<\/span>2<\/span>),<\/span> show<\/span> =<\/span> False<\/span>)<\/span>\r\n\r\nax<\/span> =<\/span> p<\/span>.<\/span>ax<\/span>\r\n\r\n# y \u8ef8\u306e\u76ee\u76db<\/span>\r\nPi<\/span> =<\/span> float<\/span>(<\/span>pi<\/span>)<\/span>\r\nax<\/span>.<\/span>set_yticks<\/span>(<\/span>\r\n    [(<\/span>0.25<\/span>*<\/span>i<\/span>*<\/span>Pi<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>2<\/span>,<\/span> 3<\/span>)])<\/span>\r\n# y \u8ef8\u306e\u30e9\u30d9\u30eb<\/span>\r\nax<\/span>.<\/span>set_yticklabels<\/span>(<\/span>\r\n    [<\/span>\"<\/span>%.2f<\/span> $\\pi$\"<\/span> %<\/span> (<\/span>0.25<\/span>*<\/span>i<\/span>)<\/span> for<\/span> i<\/span> in<\/span> range<\/span>(<\/span>-<\/span>2<\/span>,<\/span> 3<\/span>)]);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u53cc\u66f2\u7dda\u95a2\u6570<\/h3>\n
      \u5b9a\u7fa9<\/h4>\n
      \u53cc\u66f2\u7dda\u95a2\u6570\u306f\u6307\u6570\u95a2\u6570\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<\/p>\n
      $\\displaystyle \\sinh x \\equiv \\frac{e^{x} – e^{-x}}{2} =$ sinh(x)<\/code><\/p>\n
      $\\displaystyle \\cosh x \\equiv \\frac{e^{x} + e^{-x}}{2} =$ cosh(x)<\/code><\/p>\n
      $\\displaystyle \\tanh x \\equiv \\frac{\\sinh x}{\\cosh x} = \\frac{e^{x} – e^{-x}}{e^{x} + e^{-x}} =$ tanh(x)<\/code><\/p>\n
      \u53cc\u66f2\u7dda\u95a2\u6570\u3092\u6307\u6570\u95a2\u6570\u3067\u66f8\u304d\u76f4\u3059\u306b\u306f rewrite()<\/code> \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[59]:<\/div>\n
      \n
      \n
      \n
      sinh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>exp<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[59]:<\/div>\n
      $\\displaystyle \\frac{e^{x}}{2} – \\frac{e^{- x}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[60]:<\/div>\n
      \n
      \n
      \n
      cosh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>exp<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[60]:<\/div>\n
      $\\displaystyle \\frac{e^{x}}{2} + \\frac{e^{- x}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[61]:<\/div>\n
      \n
      \n
      \n
      tanh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>exp<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[61]:<\/div>\n
      $\\displaystyle \\frac{e^{x} – e^{- x}}{e^{x} + e^{- x}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u6027\u8cea<\/h4>\n
      $\\cosh^2 x – \\sinh^2 x = 1$ \u3092\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[62]:<\/div>\n
      \n
      \n
      \n
      simplify<\/span>(<\/span>cosh<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span> -<\/span> sinh<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[62]:<\/div>\n
      $\\displaystyle 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u52a0\u6cd5\u5b9a\u7406\u306e\u78ba\u8a8d<\/h4>\n
      \u53cc\u66f2\u7dda\u95a2\u6570\u306b\u3082\uff0c\u4e09\u89d2\u95a2\u6570\u306e\u5834\u5408\u3068\u4f3c\u305f\u3088\u3046\u306a\u52a0\u6cd5\u5b9a\u7406\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[63]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>expand<\/span>(<\/span>cosh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>)),<\/span> \r\n   expand<\/span>(<\/span>cosh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>),<\/span> trig<\/span> =<\/span> True<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[63]:<\/div>\n
      $\\displaystyle \\cosh{\\left(x + y \\right)} = \\sinh{\\left(x \\right)} \\sinh{\\left(y \\right)} + \\cosh{\\left(x \\right)} \\cosh{\\left(y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[64]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>expand<\/span>(<\/span>sinh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>)),<\/span> \r\n   expand<\/span>(<\/span>sinh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>),<\/span> trig<\/span> =<\/span> True<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[64]:<\/div>\n
      $\\displaystyle \\sinh{\\left(x + y \\right)} = \\sinh{\\left(x \\right)} \\cosh{\\left(y \\right)} + \\sinh{\\left(y \\right)} \\cosh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[65]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>expand<\/span>(<\/span>tanh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>)),<\/span> \r\n   expand<\/span>(<\/span>tanh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>),<\/span> trig<\/span> =<\/span> True<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[65]:<\/div>\n
      $\\displaystyle \\tanh{\\left(x + y \\right)} = \\frac{\\tanh{\\left(x \\right)}}{\\tanh{\\left(x \\right)} \\tanh{\\left(y \\right)} + 1} + \\frac{\\tanh{\\left(y \\right)}}{\\tanh{\\left(x \\right)} \\tanh{\\left(y \\right)} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u4e0a\u5f0f\u306e\u53f3\u8fba\u3092\u901a\u5206\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[66]:<\/div>\n
      \n
      \n
      \n
      expand<\/span>(<\/span>tanh<\/span>(<\/span>x<\/span> +<\/span> y<\/span>),<\/span> trig<\/span> =<\/span> True<\/span>)<\/span>.<\/span>factor<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[66]:<\/div>\n
      $\\displaystyle \\frac{\\tanh{\\left(x \\right)} + \\tanh{\\left(y \\right)}}{\\tanh{\\left(x \\right)} \\tanh{\\left(y \\right)} + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u5fae\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[67]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>sinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>sinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[67]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\sinh{\\left(x \\right)} = \\cosh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[68]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>cosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>cosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[68]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\cosh{\\left(x \\right)} = \\sinh{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[69]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>tanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   diff<\/span>(<\/span>tanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[69]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\tanh{\\left(x \\right)} = 1 – \\tanh^{2}{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u4e0a\u5f0f\u306e\u53f3\u8fba\u3092\u300c\u7c21\u5358\u5316\u300d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[70]:<\/div>\n
      \n
      \n
      \n
      diff<\/span>(<\/span>tanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[70]:<\/div>\n
      $\\displaystyle \\frac{1}{\\cosh^{2}{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u53c2\u8003\uff1a\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n
      $y = \\sinh x, \\ \\cosh x$ \u306e\u30b0\u30e9\u30d5\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[71]:<\/div>\n
      \n
      \n
      \n
      plot<\/span>((<\/span>sinh<\/span>(<\/span>x<\/span>),<\/span> \"sinh $x$\"<\/span>),<\/span> (<\/span>cosh<\/span>(<\/span>x<\/span>),<\/span> \"cosh $x$\"<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>3<\/span>,<\/span> 3<\/span>),<\/span> \r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>3<\/span>,<\/span> 3<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      $ y = \\tanh x$ \u306e\u30b0\u30e9\u30d5\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[72]:<\/div>\n
      \n
      \n
      \n
      plot<\/span>((<\/span>tanh<\/span>(<\/span>x<\/span>),<\/span> \"tanh $x$\"<\/span>),<\/span> (<\/span>x<\/span>,<\/span> -<\/span>5<\/span>,<\/span> 5<\/span>),<\/span> legend<\/span> =<\/span> True<\/span>,<\/span>\r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>5<\/span>,<\/span> 5<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u9006\u53cc\u66f2\u7dda\u95a2\u6570<\/h3>\n
      \u5b9a\u7fa9<\/h4>\n
      \u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u9006\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\uff0c\u305d\u308c\u305e\u308c<\/p>\n
      $\\sinh^{-1} x = \\mbox{arsinh}\\ x = $ asinh(x)<\/code>
      \n$\\cosh^{-1} x = \\mbox{arcosh}\\ x = $ acosh(x)<\/code>
      \n$\\tanh^{-1} x = \\mbox{artanh}\\ x = $ atanh(x)<\/code><\/p>\n
      \u3068\u66f8\u304d\u307e\u3059\u3002\uff08ar \u3067\u3042\u308a\uff0carc \u3067\u306f\u306a\u3044\u3002\uff09\u9006\u95a2\u6570\u3067\u3059\u304b\u3089\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3064\u306f\u305a\u3067\u3059\u3002<\/p>\n
      $\\sinh\\left(\\sinh^{-1} x\\right) = x$
      \n$\\cosh\\left(\\cosh^{-1} x\\right) = x$
      \n$\\tanh\\left(\\tanh^{-1} x\\right) = x$<\/p>\n
      \u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[73]:<\/div>\n
      \n
      \n
      \n
      sinh<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>)),<\/span> cosh<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>)),<\/span> tanh<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[73]:<\/div>\n
      \n
      (x, x, x)<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u53cc\u66f2\u7dda\u95a2\u6570\u306f\u6307\u6570\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u304c\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u5bfe\u6570\u95a2\u6570\u3092\u4f7f\u3063\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[74]:<\/div>\n
      \n
      \n
      \n
      asinh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[74]:<\/div>\n
      $\\displaystyle \\log{\\left(x + \\sqrt{x^{2} + 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[75]:<\/div>\n
      \n
      \n
      \n
      acosh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[75]:<\/div>\n
      $\\displaystyle \\log{\\left(x + \\sqrt{x – 1} \\sqrt{x + 1} \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[76]:<\/div>\n
      \n
      \n
      \n
      atanh<\/span>(<\/span>x<\/span>)<\/span>.<\/span>rewrite<\/span>(<\/span>log<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[76]:<\/div>\n
      $\\displaystyle – \\frac{\\log{\\left(1 – x \\right)}}{2} + \\frac{\\log{\\left(x + 1 \\right)}}{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u5fae\u5206<\/h4>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[77]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[77]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\operatorname{asinh}{\\left(x \\right)} = \\frac{1}{\\sqrt{x^{2} + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[78]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[78]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\operatorname{acosh}{\\left(x \\right)} = \\frac{1}{\\sqrt{x – 1} \\sqrt{x + 1}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[79]:<\/div>\n
      \n
      \n
      \n
      Eq<\/span>(<\/span>Derivative<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> x<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      Out[79]:<\/div>\n
      $\\displaystyle \\frac{d}{d x} \\operatorname{atanh}{\\left(x \\right)} = \\frac{1}{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      \u53c2\u8003\uff1a\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n
      $y = \\sinh^{-1} x$ \u306e\u5b9a\u7fa9\u57df\u306f $-\\infty < x < \\infty$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[80]:<\/div>\n
      \n
      \n
      \n
      plot<\/span>(<\/span>asinh<\/span>(<\/span>x<\/span>),<\/span> \"sinh$^{-1}\\, x$\"<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>10<\/span>,<\/span> 10<\/span>),<\/span> legend<\/span> =<\/span> True<\/span>)<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      $y = \\cosh^{-1} x$ \u306e\u5b9a\u7fa9\u57df\u306f $x \\geq 1$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[81]:<\/div>\n
      \n
      \n
      \n
      plot<\/span>(<\/span>acosh<\/span>(<\/span>x<\/span>),<\/span> \"cosh$^{-1}\\, x$\"<\/span>,<\/span> (<\/span>x<\/span>,<\/span> 1<\/span>,<\/span> 10<\/span>),<\/span> legend<\/span> =<\/span> True<\/span>,<\/span> \r\n     xlim<\/span> =<\/span> (<\/span>1<\/span>,<\/span> 10<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      <\/div>\n
      \n
      \n
      $y = \\tanh^{-1} x$ \u306e\u5b9a\u7fa9\u57df\u306f $-1 < x < 1$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      In\u00a0[82]:<\/div>\n
      \n
      \n
      \n
      plot<\/span>(<\/span>atanh<\/span>(<\/span>x<\/span>),<\/span> \"tanh$^{-1}\\, x$\"<\/span>,<\/span> (<\/span>x<\/span>,<\/span> -<\/span>0.999<\/span>,<\/span> 0.999<\/span>),<\/span> legend<\/span> =<\/span> True<\/span>,<\/span> \r\n     xlim<\/span> =<\/span> (<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>))<\/span>;<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
      \n
      \n
      \n
      <\/div>\n
      \n
      <\/p>\n
      \"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6034,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6036","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6036","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6036"}],"version-history":[{"count":12,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6036\/revisions"}],"predecessor-version":[{"id":8088,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6036\/revisions\/8088"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6034"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}