{"id":138,"date":"2025-09-01T00:48:13","date_gmt":"2025-08-31T15:48:13","guid":{"rendered":"https:\/\/www.masmath.online\/?p=138"},"modified":"2025-09-01T01:30:00","modified_gmt":"2025-08-31T16:30:00","slug":"%e5%90%88%e6%88%90%e9%96%a2%e6%95%b0%e3%81%ae%e5%b0%8e%e9%96%a2%e6%95%b0","status":"publish","type":"post","link":"https:\/\/www.masmath.online\/?p=138","title":{"rendered":"\u5408\u6210\u95a2\u6570\u306e\u5c0e\u95a2\u6570"},"content":{"rendered":"\n
\n

\u3000\u6b21\u306e\u5c0e\u95a2\u6570\u3092\u5fae\u5206\u305b\u3088\uff0e
\\( (1) \\)\u3000\\( y = \\sqrt{x^2 + 1} \\)

\\( (2) \\)\u3000\\( y = \\dfrac{1}{\\tan x} \\)

\\( (3) \\)\u3000\\( y = \\sin^2 2x \\)
\\( (4) \\)\u3000\\( y = e^{3x} \\cos 4x \\)<\/p>\n<\/div>\n\n\n\n

\n

\uff0a\u5fae\u5206\u516c\u5f0f\uff0a
\\( 1^\\circ \\)[\u5fae\u5206\u306e\u7dda\u5f62\u6027]\u3000\\( \\{ k f(x) + l g(x) \\}’ = k f'(x) + l g'(x) \\)
\\( 2^\\circ \\)[\u7a4d\u306e\u5fae\u5206\u516c\u5f0f]\\( \\{ f(x) g(x) \\}’ = f'(x) g(x) + f(x) g'(x) \\)
\\( 3^\\circ \\)[\u5546\u306e\u5fae\u5206\u516c\u5f0f]\u3000\\( \\left\\{ \\dfrac{f(x)}{g(x)} \\right\\}’ = \\dfrac{f'(x) g(x) – f(x) g'(x)}{\\{ g(x) \\}^2} \\)
\\( 4^\\circ \\)[ \\( x \\) \u306e\u7d2f\u4e57]\u3000\\( (x^n)’ = nx^{n-1} \\)
\\( 5^\\circ \\)[\u4e09\u89d2\u95a2\u6570]\u3000\\( (\\sin x)’ = \\cos x \\) \uff0c \\( (\\cos x)’ = – \\sin x \\) \uff0c \\( (\\tan x)’ = \\dfrac{1}{\\cos^2} \\)
\\( 6^\\circ \\)[\u6307\u6570\u95a2\u6570]\u3000\\( (e^x)’ = e^x \\) \uff0c \\( (a^x)’ = a^x \\log a \\)
\\( 7^\\circ \\)[\u5bfe\u6570\u95a2\u6570]\u3000\\( (\\log x)’ = \\dfrac{1}{x} \\) \uff0c \\( (\\log_a x)’ = \\dfrac{1}{x \\log a} \\)<\/p>\n\n\n\n

\uff0a\u5408\u6210\u95a2\u6570\u306e\u5fae\u5206\u6cd5\uff0a
\u3000\u3000\u3000\\( \\{ g(f(x)) \\}’ = g'(f(x)) \\cdot f'(x) \\)
\u3000\u3000\u3000\\( \\{ h(g(f(x))) \\}’ = h'(g(f(x))) \\cdot g'(f(x)) \\cdot f'(x) \\)<\/p>\n<\/div>\n\n\n\n

\\( (1) \\)
\u3000\\( y = (x^2 + 1)^{\\frac{1}{2}} \\) \u3067\u3042\u308b\u304b\u3089

\u3000\u3000\u3000\\( y’ = \\dfrac{1}{2} (x^2+1)^{-\\frac{1}{2}} \\cdot (x^2+1)’ \\)

\u3000\u3000\u3000\\( = \\dfrac{1}{2\\sqrt{x^2 + 1}} \\cdot 2x \\)

\u3000\u3000\u3000\\( = \\dfrac{x}{\\sqrt{x^2 + 1}} \\cdots \\) \u7b54<\/p>\n\n\n\n

\\( (2) \\)
\u3000\\( y = (\\tan x)^{-1} \\) \u3067\u3042\u308b\u304b\u3089
\u3000\u3000\u3000\\( y’ = -(\\tan x)^{-2} \\cdot (\\tan x)’ \\)

\u3000\u3000\u3000\\( = -\\dfrac{1}{\\tan^2 x} \\cdot \\dfrac{1}{\\cos^2 x} \\)

\u3000\u3000\u3000\\( = -\\dfrac{1}{\\sin^2 x} \\cdots \\) \u7b54<\/p>\n\n\n\n

\\( (3) \\)
\\( y = (\\sin 2x)^2 \\) \u3067\u3042\u308b\u304b\u3089
\u3000\u3000\u3000\\( y’ = 2 \\sin 2x \\cdot (\\sin 2x)’ \\)
\u3000\u3000\u3000\\( = 2 \\sin 2x \\cdot \\cos 2x \\cdot (2x)’ \\)
\u3000\u3000\u3000\\( = 4 \\sin 2x \\cos 2x \\cdots \\) \u7b54<\/p>\n\n\n\n

\\( (4) \\)
\u3000\u7a4d\u306e\u5fae\u5206\u516c\u5f0f\u306b\u3088\u308a
\u3000\u3000\u3000\\( y’ = (e^{3x})’ \\cos 4x + e^{3x} (\\cos 4x)’ \\)
\u3000\u3000\u3000\\( = \\{ e^{3x} \\cdot (3x)’ \\} \\cos 4x + e^{3x} \\{ (-\\sin 4x) \\cdot (4x)’ \\} \\)
\u3000\u3000\u3000\\( = 3 e^{3x} \\cos 4x – 4 e^{3x} \\sin 4x \\cdots \\) \u7b54<\/p>\n","protected":false},"excerpt":{"rendered":"

\\( (1) \\)\u3000\\( y = (x^2 + 1)^{\\frac{1}{2}} \\) \u3067\u3042\u308b\u304b\u3089 \u3000\u3000\u3000\\( y#8217; = \\dfrac{1}{2} (x^2+1)^{-\\frac{1}{2}} \\cdot (<\/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-138","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\/138","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=138"}],"version-history":[{"count":15,"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/posts\/138\/revisions"}],"predecessor-version":[{"id":163,"href":"https:\/\/www.masmath.online\/index.php?rest_route=\/wp\/v2\/posts\/138\/revisions\/163"}],"wp:attachment":[{"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.masmath.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}