{"id":2093,"date":"2022-02-22T13:01:19","date_gmt":"2022-02-22T04:01:19","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2093"},"modified":"2024-04-19T09:30:05","modified_gmt":"2024-04-19T00:30:05","slug":"%e5%af%be%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\/%e5%af%be%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u5bfe\u6570\u95a2\u6570\u306e\u5fae\u5206"},"content":{"rendered":"

\u6307\u6570\u95a2\u6570\u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u5bfe\u6570\u95a2\u6570\u306e\u3046\u3061\uff0c\u7279\u306b \\(e\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570\u3067\u3042\u308b\u300c\u81ea\u7136\u5bfe\u6570\u300d\\(\\log_e x\\) \u306e\u5fae\u5206\u306f
\n$$\\frac{d}{dx} \\left(\\log_e |x| \\right) = \\frac{1}{x} \\qquad (x \\neq 0)$$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u307e\u305f\uff0c\u5bfe\u6570\u95a2\u6570\u3092\u4f7f\u3063\u3066\uff0c\u4efb\u610f\u306e\u300c\u5b9f\u6570<\/strong><\/span>\u300d \\(r\\) \u3092\u6307\u6570\u3068\u3059\u308b\u300c\u3079\u304d\u95a2\u6570\u300d\\(x^r\\) \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3082\u8a3c\u660e\u3055\u308c\u308b\u3002<\/p>\n

$$ (x^r)’ = r\\, x^{r-1}$$<\/p>\n

\n

\u5bfe\u6570\u95a2\u6570\u3068\u306f\u6307\u6570\u95a2\u6570\u306e\u9006\u95a2\u6570\u3002<\/p>\n

\\(a\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570 \\(y = a^x \\) \u306e\u9006\u95a2\u6570\u3092\uff08\\( x \\) \u306b\u3064\u3044\u3066\u89e3\u3044\u3066\uff09\\( x = \\log_a y\\)\u00a0 \u3068\u66f8\u304d\uff0c\\( a \\) \u3068\u5e95\u3068\u3059\u308b\u5bfe\u6570\u95a2\u6570\u3068\u547c\u3076\u3002\uff08\\(y > 0 \\)\uff09<\/p>\n

\u7279\u306b\uff0c\\(e \\) \u3092\u5e95\u3068\u3059\u308b\u5bfe\u6570\u95a2\u6570\u3092\uff08\u3042\u3089\u305f\u3081\u3066 \\(x\\) \u3068 \\(y \\) \u3092\u5165\u308c\u66ff\u3048\u3066 \\(y = \\dots\\) \u306e\u5f62\u306e\u95a2\u6570\u3068\u3057\u3066\uff09
$$ y = \\log_e x \\ \\\u00a0 ( x > 0)$$\u3068\u66f8\u304d\uff0c\u300c\u81ea\u7136\u5bfe\u6570<\/strong><\/span>\u300d\u3068\u547c\u3076\u3002\u3053\u308c\u306f\uff08\\(x\\) \u3068 \\(y \\) \u3092\u5165\u308c\u66ff\u3048\u305f\u306e\u3067\uff09\\(x = e^y\\) \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u3002\u81ea\u7136\u5bfe\u6570\u306e\u5fae\u5206\u306f\uff0c\u9006\u95a2\u6570\u306e\u5fae\u5206\u6cd5\u3092\u4f7f\u3063\u3066\uff0c
$$\\frac{d}{dx} \\left(\\log_e x\\right) = \\frac{dy}{dx} = \\frac{1}{\\frac{dx}{dy}} = \\frac{1}{\\frac{d e^y}{dy}} = \\frac{1}{e^y} = \\frac{1}{x}$$<\/p>\n

\u4e0a\u8a18\u306f \\(x > 0\\) \u3067\u306e\u307f\u5b9a\u7fa9\u3055\u308c\u308b\u3002\\(x < 0\\) \u306b\u3064\u3044\u3066\u306f\uff0c\\(\\log_e (-x) \\) \u304c\u5b9a\u7fa9\u3055\u308c\uff0c\\( u = -x \\) \u3068\u3057\u3066\u5408\u6210\u95a2\u6570\u306e\u5fae\u5206\u6cd5\u3092\u4f7f\u3048\u3070
$$\\frac{d}{dx} \\left(\\log_e (-x)\\right) = \\frac{d}{du}\\left(\\log_e u\\right) \\cdot \\frac{du}{dx} = \\frac{1}{u} \\cdot (-1) = \\frac{1}{x}$$ \u307e\u3068\u3081\u308b\u3068\uff0c
$$\\frac{d}{dx} \\left(\\log_e |x|\\right) = \\frac{1}{x}\\qquad (x \\neq 0)$$<\/p>\n

\u81ea\u7136\u5bfe\u6570\u306e\u8868\u8a18<\/h4>\n

\u81ea\u7136\u5bfe\u6570\u306f\u3088\u304f\u4f7f\u308f\u308c\u308b\u306e\u3067\uff0c\u5e95 \\(e \\) \u306e\u8868\u8a18\u3092\u7701\u7565\u3057\u3066\u66f8\u304f\u3053\u3068\u3082\u3042\u308b\u3002\u307e\u305f\uff0c\u696d\u754c\u306b\u3088\u3063\u3066\u306f\u5e95\u3092\u7701\u7565\u3059\u308b\u3068\u5e38\u7528\u5bfe\u6570 \\(\\log_{10} \\) \u3067\u3042\u308b\u3068\u3059\u308b\u5834\u5408\u3082\u3042\u308b\u306e\u3067\uff0c\u306f\u3063\u304d\u308a\u533a\u5225\u3059\u308b\u305f\u3081\u306b\uff0c\\( \\ln \\) \uff08\\( n\\)\u00a0 \u306f natural \u306e \\( n\\)\uff09\u3068\u66f8\u304f\u5834\u5408\u3082\u3042\u308b\u3002\uff08\u79c1\u306f\u3053\u3061\u3089\u3092\u3088\u304f\u4f7f\u3046\u3002\uff09<\/p>\n

$$\u00a0 \\log_e x = \\log x = \\ln x$$<\/p>\n

\u3079\u304d\u95a2\u6570\u306e\u5fae\u5206\uff1a\u3075\u305f\u305f\u3073<\/h4>\n

\u6709\u7406\u6570\u3060\u3051\u3058\u3083\u306a\u304f\uff0c\u4efb\u610f\u306e\u300c\u5b9f\u6570\u300d \\(r\\) \u3092\u6307\u6570\u3068\u3059\u308b\u300c\u3079\u304d\u95a2\u6570\u300d\\( x^r\\) \u306b\u3064\u3044\u3066
$$(x^r)’ = r\\,x^{r-1}$$<\/p>\n

\u8a3c\u660e\uff1a<\/p>\n

\u4e21\u8fba\u306e\u81ea\u7136\u5bfe\u6570\u3092\u3068\u308b\u3068\uff0c$$\\log y = \\log (x^r) = r \\log x$$ \\(x\\) \u3067\u5fae\u5206\u3057\u3066
$$ \\frac{d}{dx} \\left(\\log y\\right) = \\frac{1}{y} \\frac{dy}{dx} = r \\frac{1}{x} $$
$$\\therefore\\ \\ \\frac{dy}{dx} = r \\frac{y}{x} = r \\frac{x^r}{x} = r x^{r-1}$$<\/p>\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
  3. \u4e0a\u8a18 1. \u3068 2. \u3092\u4f7f\u3046\u3068\uff0c\\(\\displaystyle \\log\\left(\\frac{x}{y}\\right) = \\log x -\\log y\\) \u3082\u7c21\u5358\u306b\u8a3c\u660e\u3067\u304d\u308b\u3002<\/li>\n<\/ol>\n

    \u4ee5\u4e0a\u306f\u7c21\u5358\u306e\u305f\u3081\u306b\u5e95\u3092 \\(e\\) \u6c7a\u3081\u6253\u3061\u306b\u3057\u3066\u66f8\u3044\u305f\u304c\uff0c\u4e00\u822c\u306e\u5e95 \\(a \\) \u3067\u3082\u540c\u69d8\u306b\u6210\u308a\u7acb\u3064\u3002<\/p>\n

    \u8a3c\u660e\u306f\uff0c\\(y = \\log x\\) \u304c \\( x = e^y \\) \u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089
    $$ x = e^y = e^{\\log x} $$ \u3067\u3042\u308b\u3053\u3068\u3092\u4f7f\u3046\u3068\uff0c
    $$ e^{\\log (x y)} = x y = e^{\\log x} \\, e^{\\log y} = e^{\\log x + \\log y}$$
    $$ e^{\\log (x^y)} = x^y = \\left( e^{\\log x} \\right)^y = e^{y \\log x}$$<\/p>\n

    <\/h4>\n

    \u5bfe\u6570\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h4>\n

    \"\"<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u6307\u6570\u95a2\u6570\u306e\u9006\u95a2\u6570\u3067\u3042\u308b\u5bfe\u6570\u95a2\u6570\u306e\u3046\u3061\uff0c\u7279\u306b \\(e\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570\u3067\u3042\u308b\u300c\u81ea\u7136\u5bfe\u6570\u300d\\(\\log_e x\\) \u306e\u5fae\u5206\u306f $$\\frac{d}{dx} \\left(\\log_e |x| \\right) = \\frac{1}{x} \\qquad (x \\neq 0)$$<\/p>\n

    \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002\u307e\u305f\uff0c\u5bfe\u6570\u95a2\u6570\u3092\u4f7f\u3063\u3066\uff0c\u4efb\u610f\u306e\u300c\u5b9f\u6570\u300d \\(r\\) \u3092\u6307\u6570\u3068\u3059\u308b\u300c\u3079\u304d\u95a2\u6570\u300d\\(x^r\\) \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3082\u8a3c\u660e\u3055\u308c\u308b\u3002<\/p>\n

    $$ (x^r)’ = r\\, x^{r-1}$$<\/p>

    \u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2093","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t