{"id":2135,"date":"2022-02-22T14:52:48","date_gmt":"2022-02-22T05:52:48","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2135"},"modified":"2024-04-17T10:27:49","modified_gmt":"2024-04-17T01:27:49","slug":"%e9%ab%98%e9%9a%8e%e5%b0%8e%e9%96%a2%e6%95%b0","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\/%e9%ab%98%e9%9a%8e%e5%b0%8e%e9%96%a2%e6%95%b0\/","title":{"rendered":"\u9ad8\u968e\u5c0e\u95a2\u6570"},"content":{"rendered":"

<\/p>\n

2\u968e\u5c0e\u95a2\u6570<\/h3>\n

\\( y = f(x)\\) \u306e\u5c0e\u95a2\u6570\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u306e\u3067\u3042\u3063\u305f\u3002
\n$$y’, \\ \\\u00a0 \\frac{dy}{dx}, \\ \\\u00a0 f'(x) , \\ \\ \u00a0 \\frac{df}{dx} $$
\n\u5c0e\u95a2\u6570\u306e\u5c0e\u95a2\u6570\u3092\u300c2\u968e\u5c0e\u95a2\u6570\u300d\u3068\u8a00\u3044\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002
\n$$y^{”}, \\ \\\u00a0 \\frac{d}{dx}\\frac{dy}{dx} = \\frac{d^2 y}{dx^2}, \\ \\\u00a0 f^{”}(x) , \\ \\ \u00a0 \\frac{d^2f}{dx^2} $$<\/p>\n

2\u968e\u5fae\u5206\u3059\u308b\u3068\u81ea\u5206\u81ea\u8eab\u306b\u30de\u30a4\u30ca\u30b9\u304c\u3064\u304f\u95a2\u6570<\/h4>\n

\u4e09\u89d2\u95a2\u6570\u30922\u968e\u5fae\u5206\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\left( \\sin x \\right)’ &=& \\cos x \\\\
\n\\left( \\sin x \\right)^{”} &=& \\left( \\cos x \\right)’ = -\\sin x
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\left( \\cos x \\right)’ &=& -\\sin x \\\\
\n\\left( \\cos x \\right)^{”} &=& -\\left( \\sin x \\right)’\u00a0 =\u00a0 -\\cos x
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a\uff0c$\\sin x, \\ \\cos x$ \u3092 $x$ \u30672\u968e\u5fae\u5206\u3059\u308b\u3068\uff0c\u81ea\u5206\u81ea\u8eab\u306b\u30de\u30a4\u30ca\u30b9\u304c\u3064\u3044\u305f\u7b54\u3048\u306b\u306a\u308b\u3002\u3053\u308c\u3092\u899a\u3048\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

2\u5e74\u751f\u306e\u300c\u7406\u5de5\u7cfb\u306e\u6570\u5b66C\u300d\u306e2\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u9805<\/a>\u3067\uff0c\u300c2\u968e\u5fae\u5206\u3057\u305f\u3089\u81ea\u5206\u81ea\u8eab\u306b\u30de\u30a4\u30ca\u30b9\u304c\u3064\u304f\u95a2\u6570\u3063\u3066\u306a\u30fc\u3093\u3060\uff1f<\/strong><\/span>\u300d\u3068\u8cea\u554f\u3059\u308b\u306e\u3067\uff0c\u305d\u306e\u3068\u304d\u306f\u810a\u9ac4\u53cd\u5c04\u3067\u300c$\\cos x$\u00a0 \u3068 $\\sin x$ \u3060\u30fc\u3063\uff01<\/strong><\/span>\u300d \u3068\u53eb\u3076\u3088\u3046\u306b\u3002<\/p>\n

2\u968e\u5fae\u5206\u3059\u308b\u3068\u81ea\u5206\u81ea\u8eab\u306b\u306a\u308b\u95a2\u6570<\/h4>\n

\u53cc\u66f2\u7dda\u95a2\u6570\u30922\u968e\u5fae\u5206\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\left( \\sinh x \\right)’ &=& \\cosh x \\\\
\n\\left( \\sinh x \\right)^{”} &=& \\left( \\cosh x \\right)’\u00a0 = \\sinh x
\n\\end{eqnarray}<\/p>\n

\\begin{eqnarray}
\n\\left( \\cosh x \\right)’ &=& \\sinh x \\\\
\n\\left( \\cosh x \\right)^{”} &=& \\left( \\sinh x \\right)’ = \\cosh x
\n\\end{eqnarray}<\/p>\n

\u3064\u307e\u308a\uff0c$\\sinh x, \\ \\cosh x$ \u3092 $x$ \u30672\u968e\u5fae\u5206\u3059\u308b\u3068\uff0c\u81ea\u5206\u81ea\u8eab\u306b\u306a\u308b\u3002\u3053\u308c\u3092\u899a\u3048\u3066\u304a\u3044\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

2\u5e74\u751f\u306e\u300c\u7406\u5de5\u7cfb\u306e\u6570\u5b66C\u300d\u306e2\u968e\u7dda\u5f62\u5e38\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u9805<\/a>\u3067\uff0c\u300c2\u968e\u5fae\u5206\u3057\u305f\u3089\u81ea\u5206\u81ea\u8eab\u306b\u306a\u308b\u95a2\u6570\u3063\u3066\u306a\u30fc\u3093\u3060\uff1f<\/strong><\/span>\u300d\u3068\u8cea\u554f\u3059\u308b\u306e\u3067\uff0c\u305d\u306e\u3068\u304d\u306f\u810a\u9ac4\u53cd\u5c04\u3067\u300c$\\cosh x$\u00a0 \u3068 $\\sinh x$ \u3060\u30fc\u3063\uff01<\/strong><\/span>\u300d \u3068\u53eb\u3076\u3088\u3046\u306b\u3002<\/p>\n

\u3042\u308b\u3044\u306f\uff0c\u6307\u6570\u95a2\u6570 $e^x$ \u3084 $e^{-x}$ \u30922\u968e\u5fae\u5206\u3059\u308b\u3068…<\/p>\n

\\begin{eqnarray}
\n\\left( e^x\\right)^{”} &=& \\left( e^x \\right)’\u00a0 = e^x \\\\
\n\\left( e^{-x}\\right)^{”} &=& -\\left( e^{-x} \\right)’\u00a0 = e^{-x}
\n\\end{eqnarray}<\/p>\n

\u306a\u306e\u3067\uff0c\u810a\u9ac4\u53cd\u5c04\u3067\u300c$e^x$\u00a0 \u3068 $e^{-x}$ \u3060\u30fc\u3063\uff01<\/strong><\/span>\u300d \u3068\u53eb\u3093\u3067\u3082\u3088\u3044\u3002<\/p>\n

\u9ad8\u968e\uff08\u4e00\u822c\u306b \\(n\\) \u968e\uff09\u5c0e\u95a2\u6570<\/h3>\n

\u9ad8\u968e\uff08\u4e00\u822c\u306b \\(n\\) \u968e\uff09\u306e\u5c0e\u95a2\u6570\u3082\u540c\u69d8\u306b\u30c0\u30c3\u30b7\u30e5<\/strong><\/span>\uff08\u307e\u305f\u306f\u6b27\u7c73\u306a\u307e\u308a\u3067\u30d7\u30e9\u30a4\u30e0<\/strong><\/span>\uff09 \\({}’\\) \u3092 \\(n\\) \u500b\u66f8\u304f\u306e\u306f\u5927\u5909\u306a\u306e\u3067\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3002<\/p>\n

$$y^{(n)}, \\ \\\u00a0 \\frac{d^n y}{dx^n}, \\ \\\u00a0 f^{(n)}(x) , \\ \\ \u00a0 \\frac{d^n f}{dx^n} $$<\/p>\n

\u53c2\u8003\uff1a\u4e00\u822c\u5316\u3055\u308c\u305f\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb<\/h3>\n

\u5fae\u5206\u6cd5\u306e\u516c\u5f0f 3. \u3067\uff0c\u95a2\u6570\u306e\u7a4d\u306b\u95a2\u3059\u308b\u5fae\u5206\u304c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u3057\u305f\u3002<\/p>\n

$$\\left\\{ f(x)\\, g(x) \\right\\}’ = f'(x)\\, g(x) + f(x)\\, g'(x)$$<\/p>\n

2\u968e\u5fae\u5206\u306f\uff0c
\n\\begin{eqnarray}
\n\\left\\{ f(x)\\, g(x) \\right\\}^{”} &=& \\left\\{ f'(x)\\, g(x) + f(x)\\, g'(x) \\right\\}’\\\\
\n&=& f^{”}(x) g(x) + f'(x) g'(x) + f'(x) g'(x) + f(x) g^{”}(x) \\\\
\n&=& f^{”}(x) g(x) + 2 f'(x) g'(x) + f(x) g^{”}(x)
\n\\end{eqnarray}<\/p>\n

3\u968e\u5fae\u5206\u306f\uff0c
\n\\begin{eqnarray}
\n\\left\\{ f(x)\\, g(x) \\right\\}^{\\prime\\prime\\prime} &=& f^{\\prime\\prime\\prime}(x) g(x) + 3 f^{\\prime\\prime}(x) g'(x) + 3 f'(x) g^{\\prime\\prime}(x) +f(x) g^{\\prime\\prime\\prime}(x)
\n\\end{eqnarray}<\/p>\n

\u305d\u308c\u305e\u308c\u306e\u9805\u306e\u524d\u306e\u7cfb\u6570\u306f \\( (a+b)^2 \\) \u3084 \\( (a+b)^3 \\) \u306e\u5404\u9805\u306e\u7cfb\u6570\u3068\u540c\u3058\u300c2\u9805\u4fc2\u6570\u300d\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u3053\u306e\u3053\u3068\u3092\u4e00\u822c\u5316\u3055\u308c\u305f\u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u30eb\u30fc\u30eb\u3068\u547c\u3093\u3067\u3044\u308b\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":11,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2135","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\/2135","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=2135"}],"version-history":[{"count":11,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2135\/revisions"}],"predecessor-version":[{"id":8424,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2135\/revisions\/8424"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2068"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2135"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}