{"id":2131,"date":"2022-02-22T14:22:41","date_gmt":"2022-02-22T05:22:41","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2131"},"modified":"2024-04-17T14:55:46","modified_gmt":"2024-04-17T05:55:46","slug":"%e3%81%be%e3%81%a8%e3%82%81%ef%bc%9a%e5%88%9d%e7%ad%89%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\/%e3%81%be%e3%81%a8%e3%82%81%ef%bc%9a%e5%88%9d%e7%ad%89%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u307e\u3068\u3081\uff1a\u521d\u7b49\u95a2\u6570\u306e\u5fae\u5206"},"content":{"rendered":"<p><!--more--><\/p>\n<h3>\u3079\u304d\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u4efb\u610f\u306e\u5b9f\u6570 $r$ \u3092\u6307\u6570\u3068\u3059\u308b\u300c\u3079\u304d\u95a2\u6570\u300d\\(x^r\\) \u306b\u3064\u3044\u3066\uff0c<br \/>\n$$(x^r)&#8217; = r\\,x^{r-1}$$<\/p>\n<h3>\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u30cd\u30a4\u30d4\u30a2\u6570 \\(e\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570 \\( y = e^x \\) \u306e\u5fae\u5206\u306f $$\\left(e^x\\right)&#8217; = e^x $$<\/p>\n<h3>\u5bfe\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u300c\u81ea\u7136\u5bfe\u6570\u300d\\(\\log_e x\\) \u306e\u5fae\u5206\u306f<br \/>\n$$\\frac{d}{dx} \\left(\\log_e |x| \\right) = \\frac{1}{x} \\qquad (x \\neq 0)$$<\/p>\n<p>\u81ea\u7136\u5bfe\u6570\u306e\u8868\u8a18\u306f\u5e95\u3092\u7701\u7565\u3057\u3066 \\(\\log x\\) \u3068\u66f8\u3044\u305f\u308a\uff0c\u300c\u81ea\u7136\u300d\u5bfe\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u306f\u3063\u304d\u308a\u3055\u305b\u308b\u305f\u3081\u306b \\(\\ln x\\) \u3068\u66f8\u3044\u305f\u308a\u3059\u308b\u3002<\/p>\n<h3>\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u5f27\u5ea6\u6cd5\uff08\u30e9\u30b8\u30a2\u30f3\u5358\u4f4d\uff09\u3067 \\(x\\) \u3092\u8868\u3059\u3068\uff0c<br \/>\n$$(\\sin x)&#8217; = \\cos x, \\quad (\\cos x)&#8217; = -\\sin x, \\quad (\\tan x)&#8217; = \\frac{1}{\\cos^2 x}$$<\/p>\n<h3>\u9006\u4e09\u89d2\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u9006\u4e09\u89d2\u95a2\u6570 \\(\\sin^{-1} x , \\cos^{-1} x, \\tan^{-1} x \\) \u306e\u5fae\u5206\u3002<br \/>\n$$\\left(\\sin^{-1} x\\right)&#8217; =\u00a0 \\frac{1}{\\sqrt{1-x^2}}, \\quad \\left( \\cos^{-1} x \\right)&#8217;= -\\frac{1}{\\sqrt{1-x^2}}, \\quad \\left( \\tan^{-1} x \\right)&#8217;= \\frac{1}{1 + x^2}$$<\/p>\n<p>\u9006\u4e09\u89d2\u95a2\u6570\u306f\u4e09\u89d2\u95a2\u6570\u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u3002\u5b9a\u7fa9\u3092\u3057\u3063\u304b\u308a\u7406\u89e3\u3059\u308b\u3053\u3068\u3002<\/p>\n<h3>\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u4e09\u89d2\u95a2\u6570\u3068\u7d1b\u3089\u308f\u3057\u3044\u8868\u8a18\u306e\u300c\u53cc\u66f2\u7dda\u95a2\u6570\u300d\u306e\u5b9a\u7fa9\u3002<br \/>\n$$\\cosh x \\equiv \\frac{e^x + e^{-x}}{2}, \\quad \\sinh x \\equiv \\frac{e^x -e^{-x}}{2}, \\quad \\tanh x \\equiv \\frac{\\sinh x}{\\cosh x}$$ \u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u306f<\/p>\n<p>$$(\\cosh x)&#8217; = \\sinh x, \\quad (\\sinh x)&#8217; = \\cosh x, \\quad (\\tanh x)&#8217; = \\frac{1}{\\cosh^2 x}$$<\/p>\n<h3>\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n<p>\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u5fae\u5206\u3002\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u8868\u8a18\u306f\uff0c<\/p>\n<p>$$ \\cosh^{-1} x, \\quad \\sinh^{-1} x, \\quad \\tanh^{-1} x$$<\/p>\n<p>\u53cc\u66f2\u7dda\u95a2\u6570\u306f\u6307\u6570\u95a2\u6570 \\(e^x\\) \u3092\u4f7f\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u305f\u306e\u3067\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u300c\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u300d\u306f\u6307\u6570\u95a2\u6570\u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u81ea\u7136\u5bfe\u6570\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3082\u8868\u3055\u308c\u308b\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\cosh^{-1} x &amp;=&amp; \\log\\left( x + \\sqrt{x^2 -1}\\right) \\\\<br \/>\n\\sinh^{-1} x &amp;=&amp; \\log\\left( x + \\sqrt{x^2 +1}\\right) \\\\<br \/>\n\\tanh^{-1} x &amp;=&amp; \\frac{1}{2} \\log\\left(\\frac{1+x}{1-x}\\right)<br \/>\n\\end{eqnarray}<\/p>\n<p>\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u306f<\/p>\n<p>&nbsp;<\/p>\n<p>$$(\\cosh^{-1} x)&#8217; =\u00a0 \\frac{1}{\\sqrt{x^2-1}}, \\quad(\\sinh^{-1} x)&#8217; =\u00a0 \\frac{1}{\\sqrt{x^2 + 1}}, \\quad (\\tanh^{-1} x)&#8217; =\u00a0 \\frac{1}{1-x^2}$$<\/p>\n<p>\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u3068\u306f\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u9006\u95a2\u6570\u3002\u5b9a\u7fa9\u3092\u3057\u3063\u304b\u308a\u7406\u89e3\u3059\u308b\u3053\u3068\u3002<\/p>\n<p>\u3061\u306a\u307f\u306b\uff0c\u53cc\u66f2\u7dda\u95a2\u6570\u3060\u3051\u3067\u3082\u3042\u307e\u308a\u898b\u304b\u3051\u306a\u3044\u306e\u306b\uff0c\u3055\u3089\u306b\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306a\u3093\u304b\u3044\u3063\u305f\u3044\u3044\u3064\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\u3068\u601d\u3046\u304b\u3082\u3057\u308c\u306a\u3044\u3002\u4ee5\u4e0b\u306e\u30da\u30fc\u30b8\u3092\u53c2\u7167\u3002\u3053\u3093\u306a\u3068\u3053\u308d\u3067\uff0c\u5b87\u5b99\u5e74\u9f62\u304c 138\u5104\u5e74\u3067\u3042\u308b\u3053\u3068\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\uff0c\u9006\u53cc\u66f2\u7dda\u95a2\u6570 $\\tanh^{-1} x$ \u304c\u4f7f\u308f\u308c\u308b\u3093\u3067\u3059\u3088\u3002<\/p>\n<ul>\n<li><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/4362\/\">Planck 2018 results \u304b\u3089 Maxima \u3067\u5b87\u5b99\u5e74\u9f62\u3092\u8a08\u7b97\u3059\u308b<\/a><\/li>\n<\/ul>\n<p>\u5b87\u5b99\u5e74\u9f62 $t_0$ \u306f<\/p>\n<p>$$t_0 = \\frac{1}{H_0} \\times \\frac{2}{3(\\sqrt{1-\\Omega_{\\rm m} })}\\tanh^{-1} \\sqrt{1-\\Omega_{\\rm m} }$$<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2131","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.0.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u3079\u304d\u95a2\u6570\u306e\u5fae\u5206 \u4efb\u610f\u306e\u5b9f\u6570 $r$ \u3092\u6307\u6570\u3068\u3059\u308b\u300c\u3079\u304d\u95a2\u6570\u300d\\(x^r\\) 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