{"id":97,"date":"2025-08-31T03:24:41","date_gmt":"2025-08-30T18:24:41","guid":{"rendered":"https:\/\/www.masmath.online\/?p=97"},"modified":"2025-09-01T01:29:43","modified_gmt":"2025-08-31T16:29:43","slug":"%e5%b0%8e%e9%96%a2%e6%95%b0%e3%81%ae%e5%ae%9a%e7%be%a9%e3%81%a8%e8%a8%88%e7%ae%97","status":"publish","type":"post","link":"https:\/\/www.masmath.online\/?p=97","title":{"rendered":"\u5c0e\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u8a08\u7b97"},"content":{"rendered":"\n
\n

\u3000\u5b9a\u7fa9\u306b\u5f93\u3063\u3066\uff0c\u6b21\u306e\u95a2\u6570\u3092\u5fae\u5206\u305b\u3088\uff0e
\\( (1) \\)\u3000\\( f(x) = x^n \\)
\\( (2) \\)\u3000\\( f(x) = \\sin x \\)
\\( (3) \\)\u3000\\( f(x) = e^x \\)
\\( (4) \\)\u3000\\( f(x) = \\log x \\)<\/p>\n<\/div>\n\n\n\n

\n

\uff0a\u5c0e\u95a2\u6570\uff0a
\u3000\u95a2\u6570 \\( f(x) \\) \u306e\u5c0e\u95a2\u6570\u306f
\u3000\u3000\u3000\\( f'(x) = \\displaystyle\\lim_{h \\to 0} \\dfrac{f(x+h)-f(x)}{h} \\)<\/p>\n<\/div>\n\n\n\n

\\( (1) \\)
\u3000\u3000\u3000\\( f'(x) = \\displaystyle\\lim_{h \\to 0} \\dfrac{(x+h)^n – x^n}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\dfrac{n x^{n-1} h + {}_n \\mathrm{C}_2 x^{n-2} h^2 + \\cdots + h^n}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} (n x^{n-1} + {}_n \\mathrm{C}_2 x^{n-2} h + \\cdots + h^{n-1}) \\)

\u3000\u3000\u3000\\( = n x^{n-1} \\cdots \\) \u7b54<\/p>\n\n\n\n

\\( (2) \\)
\u3000\u3000\u3000\\( f'(x) = \\displaystyle\\lim_{h \\to 0} \\dfrac{\\sin (x+h) – \\sin x}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\dfrac{\\sin x \\cos h + \\cos x \\sin h – \\sin x}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\dfrac{\\sin x (\\cos h – 1) + \\cos x \\sin h}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\left( \\dfrac{(\\cos h – 1)(\\cos h + 1)}{\\cos h + 1} \\cdot \\sin x + \\dfrac{\\sin h}{h} \\cdot \\cos x \\right) \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\left( \\dfrac{- \\sin^2 h}{\\cos h + 1} \\cdot \\sin x + \\dfrac{\\sin h}{h} \\cdot \\cos x \\right) \\)

\u3000\u3000\u3000\\( = \\dfrac{0}{1+1} \\cdot \\sin x + 1 \\cdot \\cos x \\)

\u3000\u3000\u3000\\( = \\cos x \\cdots \\) \u7b54<\/p>\n\n\n\n

\\( (3) \\)
\u3000\u3000\u3000\\( f'(x) = \\displaystyle\\lim_{h \\to 0} \\dfrac{e^{x+h} – e^x}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\dfrac{e^x (e^h – 1)}{h} \\)

\u3000\u3000\u3000\\( = e^x \\cdot 1 \\)

\u3000\u3000\u3000\\( = e^x \\cdots \\) \u7b54<\/p>\n\n\n\n

\\( (4) \\)
\u3000\u3000\u3000\\( f'(x) = \\displaystyle\\lim_{h \\to 0} \\dfrac{\\log (x+h) – \\log x}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\dfrac{\\log \\left( 1 + \\dfrac{h}{x} \\right)}{h} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\dfrac{x}{h} \\log \\left( 1 + \\dfrac{h}{x} \\right) \\cdot \\dfrac{1}{x} \\)

\u3000\u3000\u3000\\( = \\displaystyle\\lim_{h \\to 0} \\log \\left( 1 + \\dfrac{h}{x} \\right)^{\\frac{x}{h}} \\cdot \\dfrac{1}{x} \\)

\u3000\u3000\u3000\\( = (\\log e) \\cdot \\dfrac{1}{x} \\)

\u3000\u3000\u3000\\( = \\dfrac{1}{x} \\cdots \\) \u7b54<\/p>\n\n\n\n

\n

\\( 1^\\circ \\)
\u3000\\( (2) \\) \u3067\u306f
\u3000\u3000\u3000\\( \\displaystyle\\lim_{h \\to 0} \\dfrac{\\sin h}{h} = 1 \\) \uff0c
\\( (3) \\) \u3067\u306f
\u3000\u3000\u3000\\( \\displaystyle\\lim_{h \\to 0} \\dfrac{e^h – 1}{h} = 1 \\) \uff0c
\\( (4) \\) \u3067\u306f
\u3000\u3000\u3000\\( \\displaystyle\\lim_{t \\to \\infty} \\left( 1 + \\dfrac{1}{t} \\right)^t = \\displaystyle\\lim_{h \\to 0} (1+h)^{\\frac{1}{h}} = e \\)
\u3068\u306a\u308b\u3053\u3068\u3092\u305d\u308c\u305e\u308c\u7528\u3044\u3066\u3044\u308b\uff0e<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\\( (1) \\)\u3000\u3000\u3000\\( f'(x) = \\displaystyle\\lim_{h \\to 0} \\dfrac{(x+h)^n #8211; x^n}{h} \\) \u3000\u3000\u3000\\( = \\displaystyle\\lim<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jin_ogp_image_url":"","_jin_last_featured_id":0,"footnotes":""},"categories":[4],"tags":[26,33],"class_list":["post-97","post","type-post","status-publish","format-standard","hentry","category-essence-course","tag-differential-method-math-iii","tag-analysis"],"_links":{"self":[{"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/posts\/97","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=97"}],"version-history":[{"count":31,"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/posts\/97\/revisions"}],"predecessor-version":[{"id":162,"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/posts\/97\/revisions\/162"}],"wp:attachment":[{"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=97"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=97"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=97"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}