{"id":2084,"date":"2022-02-22T12:31:00","date_gmt":"2022-02-22T03:31:00","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2084"},"modified":"2024-04-17T10:21:45","modified_gmt":"2024-04-17T01:21:45","slug":"%e6%8c%87%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\/%e6%8c%87%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206"},"content":{"rendered":"

\\(a \\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570 \\( y = a^x \\) \u306e\u3046\u3061\uff0c\u7279\u306b\u30cd\u30a4\u30d4\u30a2\u6570<\/strong><\/span> \\(e\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570 \\( y = e^x \\) \u306e\u5fae\u5206\u306f $$\\left(e^x\\right)’ = e^x $$<\/p>\n

\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>\n

\n

\\( y = a^x\\) \u306e\u5fae\u5206\u306f\u5b9a\u7fa9\u304b\u3089
$$ \\frac{dy}{dx} = \\lim_{h \\rightarrow 0} \\frac{a^{x + h} -a^x}{h} = a^x \\lim_{h \\rightarrow 0} \\frac{a^{h} -1}{h}$$<\/p>\n

\u30cd\u30a4\u30d4\u30a2\u6570\u306e\u5b9a\u7fa9<\/h3>\n

\u6307\u6570\u95a2\u6570\u306e\\(x = 0\\) \u306b\u5fae\u5206\u7cfb\u6570\uff08\u5c0e\u95a2\u6570\u306e\u5024\uff09\u304c\u7279\u306b \\(1\\)\uff0c\u3059\u306a\u308f\u3061\uff08\\(a^0 = 1\\) \u3067\u3042\u308b\u304b\u3089\uff09
$$\\lim_{h \\rightarrow 0} \\frac{a^{h} -1}{h} = 1$$
\u306b\u306a\u308b\u3088\u3046\u306a\u5e95\\(a\\) \u3092\u7279\u306b \\(e\\) \u3068\u66f8\u304d\uff0c\u300c\u30cd\u30a4\u30d4\u30a2\u6570<\/strong><\/span>\u300d\u3068\u547c\u3076\u3002\uff08\u3053\u306e\u547c\u3073\u65b9\u306f\u6570\u5b66\u7551\u3067\u306a\u3044\u79c1\u306b\u3068\u3063\u3066\u306f\u3042\u307e\u308a\u4e00\u822c\u7684\u3067\u306f\u306a\u3044\u304b\u306a\u3041\u300220\u4e16\u7d00\u306e\u5927\u6614\u306e\u9ad8\u6821\u3067\u3082\u3053\u306e\u540d\u524d\u3092\u805e\u3044\u305f\u3053\u3068\u306a\u3044\u3057\u3002\u3042\u3068\u3067\u81ea\u7136\u5bfe\u6570\u306e\u3068\u3053\u308d\u3067\uff0c\\(e\\) \u3092\u81ea\u7136\u5bfe\u6570\u306e\u5e95<\/strong><\/span>\u3068\u547c\u3076\u304c\u3053\u3061\u3089\u306e\u307b\u3046\u304c\uff0c\u99b4\u67d3\u307f\u304c\u3042\u308b\u3002\uff09<\/p>\n

\u30cd\u30a4\u30d4\u30a2\u6570\u306e\u5b9a\u7fa9\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u308b\u3002$h \\rightarrow 0$ \u306e\u6975\u9650\u3067 $1$ \u3068\u3044\u3046\u3053\u3068\u306f\uff0c\u6975\u9650\u3092\u53d6\u308b\u524d\u306f $1 + O(h)$ \u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3059\u308b\u3002\u3053\u3053\u3067\uff0c$O(h)$ \u306f $h$ \u306e1\u4e57\u4ee5\u4e0a\u306e\u9805\u3068\u3044\u3046\u610f\u5473\u3002
\n$h \\rightarrow 0$ \u306e\u6975\u9650\u3067 $O(h) \\rightarrow 0$ \u306b\u306a\u308b\u304b\u3089\u3002
\\begin{eqnarray}
\\lim_{h \\rightarrow 0} \\frac{e^{h} -1}{h} &=& 1\\\\
\n\\frac{e^{h} -1}{h} &=& 1 + O(h)\\\\
\ne^h -1 &=& h + h \\cdot O(h) = h + O(h^2)\\\\
e^h &=& 1 + h + O(h^2)\\\\
e &=& \\left(1 + h+ O(h^2)\\right)^{\\frac{1}{h}}\\\\
\\therefore\\ e &=& \\lim_{h \\rightarrow 0} \\left(1 + h\\right)^{\\frac{1}{h}}
\\end{eqnarray}
\u3053\u3053\u3067 \\(O(h^2)\\) \u3068\u306f \\(h^2\\) \u4ee5\u4e0a\u306e\u9ad8\u6b21\u306e\u9805\u3068\u3044\u3046\u610f\u5473\u3067\u3042\u308b\u3002<\/p>\n

\u30cd\u30a4\u30d4\u30a2\u6570\u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n

\u30cd\u30a4\u30d4\u30a2\u6570 \\( e\\) \u306e\u5b9a\u7fa9\u3068\u306f\u3064\u307e\u308a\uff0c
$$ \\lim_{h \\rightarrow 0} \\frac{e^{h} -1}{h} = 1$$ \u3067\u3042\u308b\u3002\u3053\u306e \\(e\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206\u306f\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u3088\u308a\uff0c
$$\\frac{d}{dx} e^x =\u00a0 \\lim_{h \\rightarrow 0} \\frac{e^{x+h} -e^x}{h} = e^x \\lim_{h \\rightarrow 0} \\frac{e^{h} -1}{h} = e^x \\cdot 1 = e^x$$\u3059\u306a\u308f\u3061 $$\\left(e^x\\right)’\u00a0 = e^x $$<\/p>\n

\u6307\u6570\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h3>\n

\"\"<\/p>\n

\u53c2\u8003\uff1a\u6307\u6570\u6cd5\u5247<\/h3>\n
    \n
  1. \\(a^x\\,a^y = a^{x+y}\\)<\/li>\n
  2. \\(\\displaystyle \\left(a^x\\right)^y = a^{xy} \\)<\/li>\n
  3. \\( (a b)^x = a^x\\, b^x\\)<\/li>\n<\/ol>\n

    <\/h4>\n","protected":false},"excerpt":{"rendered":"

    \\(a \\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570 \\( y = a^x \\) \u306e\u3046\u3061\uff0c\u7279\u306b\u30cd\u30a4\u30d4\u30a2\u6570 \\(e\\) \u3092\u5e95\u3068\u3059\u308b\u6307\u6570\u95a2\u6570 \\( y = e^x \\) \u306e\u5fae\u5206\u306f $$\\left(e^x\\right)’ = e^x $$<\/p>\n

    \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059\u3002<\/p>

    \u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":2068,"menu_order":4,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2084","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