{"id":6714,"date":"2021-07-21T00:03:36","date_gmt":"2021-07-20T15:03:36","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6714"},"modified":"2021-09-24T00:03:25","modified_gmt":"2021-09-23T15:03:25","slug":"differentiation-theorems","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/differentiation-theorems\/","title":{"rendered":"\ubbf8\ubd84 \uacf5\uc2dd"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 5\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uae30\ubcf8 \uacf5\uc2dd<\/h2>\n<p>\\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(x\\)\uac00 \uc2e4\ubcc0\uc218\uc774\uba70 \\(f(x) = x^n\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\ub9cc\uc57d \\(n=1\\)\uc774\uba74<br \/>\n\\[f &#8216; (x) = \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) &#8211; f(x)}{\\Delta x} = \\lim_{\\Delta x \\rightarrow 0} \\frac{(x+\\Delta x) -x}{\\Delta x} = 1\\]<br \/>\n\uc774\ub2e4. \ub9cc\uc57d \\(n \\ge 2\\)\uc774\uba74<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x) &#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) &#8211; f(x)}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{(x+\\Delta x)^n &#8211; x^n}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\left\\{ ( x+\\Delta x ) -x \\right\\} \\left\\{ (x+\\Delta x)^{n-1} + (x+\\Delta x)^{n-2} x + \\cdots + x^{n-1} \\right\\}}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\left\\{ (x+ \\Delta x)^{n-1} + (x+\\Delta x)^{n-2} x + (x+\\Delta x)^{n-3} x^2 + \\cdots + x^{n-1} \\right\\} \\\\[4pt]<br \/>\n&#038;=<br \/>\n\\underbrace{x^{n-1} + x^{n-1} + x^{n-1} + \\cdots + x^{n-1}}_{n\\,\\,\\text{terms}} \\\\[4pt]<br \/>\n&#038;= nx^{n-1}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc9c0\uae08\uae4c\uc9c0 \ub2e4\uc74c \uc815\ub9ac\ub97c \uc99d\uba85\ud558\uc600\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.2.1. (\uac70\ub4ed\uc81c\uacf1 \ubc95\uce59; Power Rule)<\/span><\/p>\n<p>\\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(f\\)\uac00 \\(f(x)=x^n\\)\uc73c\ub85c \uc815\uc758\ub41c \uc2e4\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(n=1\\)\uc774\uba74, \\(f &#8216; (x) = 1\\)\uc774\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(n \\ge 2\\)\uc774\uba74, \\(f &#8216; (x) = nx^{n-1}\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\ub2e4\uc74c\uc73c\ub85c \uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc0b4\ud3b4 \ubcf4\uc790. \\(k\\)\uac00 \uc0c1\uc218\uc774\uace0, \\(f\\)\uc640 \\(g\\)\uac00 \uac19\uc740 \uc815\uc758\uc5ed\uc744 \uac00\uc9c0\uba70 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<p><span class=\"definition\">[1] \\(\\boldsymbol{kf}\\)\uc758 \ub3c4\ud568\uc218.<\/span><br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\left\\{ kf(x) \\right\\}<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{kf(x+\\Delta x) &#8211; kf(x)}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= k \\times \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x)-f(x)}{\\Delta x} \\\\[6pt]<br \/>\n&#038;= kf &#8216; (x) .<br \/>\n\\end{align}\\]\n<\/p>\n<p><span class=\"definition\">[2] \\(\\boldsymbol{(f+g)}\\)\uc758 \ub3c4\ud568\uc218.<\/span><br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx}\\left\\{ f(x) + g(x) \\right\\}<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\left\\{ f(x+\\Delta x) + g(x+\\Delta x) \\right\\} &#8211; \\left\\{ f(x)+g(x)\\right\\}  }{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) &#8211; f(x)}{\\Delta x} + \\lim_{\\Delta x \\rightarrow 0} \\frac{g(x+\\Delta x) &#8211; g(x)}{\\Delta x} \\\\[6pt]<br \/>\n&#038;= f &#8216; (x) + g &#8216; (x) .<br \/>\n\\end{align}\\]\n<\/p>\n<p><span class=\"definition\">[3] \\(\\boldsymbol{(f-g)}\\)\uc758 \ub3c4\ud568\uc218.<\/span><br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\left\\{ f(x) &#8211; g(x) \\right\\}<br \/>\n&#038;= \\frac{d}{dx} \\left\\{ f(x) + (-1) \\cdot g(x) \\right\\} \\\\[4pt]<br \/>\n&#038;= \\frac{d}{dx} f(x) + \\frac{d}{dx} \\left\\{ (-1) g(x) \\right\\} \\\\[6pt]<br \/>\n&#038;= f &#8216; (x) + (-1) g &#8216; (x) = f &#8216; (x) &#8211; g &#8216; (x) .<br \/>\n\\end{align}\\]\n<\/p>\n<p><span class=\"definition\">[4] \\(\\boldsymbol{fg}\\)\uc758 \ub3c4\ud568\uc218.<\/span><br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\left\\{ f(x)g(x) \\right\\}<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x)g(x+\\Delta x) &#8211; f(x)g(x)}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{ f(x+\\Delta x)g(x+\\Delta x ) &#8211; f(x) g(x+\\Delta x) + f(x)g(x+\\Delta x) &#8211; f(x)g(x) }{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\left\\{ \\frac{f(x+\\Delta x) -f(x)}{\\Delta x} \\cdot g(x+\\Delta x) \\right\\} + \\lim_{\\Delta x \\rightarrow 0} \\left\\{ f(x) \\cdot \\frac{g(x+\\Delta x) -g(x)}{\\Delta x}\\right\\} \\\\[6pt]<br \/>\n&#038;= f &#8216; (x) g(x) + f(x) g &#8216; (x) .<br \/>\n\\end{align}\\]\n<\/p>\n<p><span class=\"definition\">[5] \\(\\boldsymbol{f\/g}\\)\uc758 \ub3c4\ud568\uc218.<\/span> \\(g(x) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790.<br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} \\left\\{ \\frac{f(x)}{g(x)} \\right\\}<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{\\displaystyle\\frac{f(x+\\Delta x)}{g(x+\\Delta x)} &#8211; \\frac{f(x)}{g(x)}}{\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x)g(x) &#8211; f(x)g(x+\\Delta x)}{g(x)g(x+\\Delta x )\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x)g(x) &#8211; f(x)g(x) &#8211; \\left\\{ f(x)g(x+\\Delta x ) &#8211; f(x)g(x) \\right\\}}{g(x)g(x+\\Delta x)\\Delta x} \\\\[4pt]<br \/>\n&#038;= \\lim_{\\Delta x \\rightarrow 0} \\frac{1}{g(x)g(x+\\Delta x)} \\left\\{ \\frac{f(x+\\Delta x) -f(x)}{\\Delta x} \\cdot g(x) &#8211; f(x) \\cdot \\frac{g(x+\\Delta x) &#8211; g(x)}{\\Delta x} \\right\\} \\\\[4pt]<br \/>\n&#038;= \\frac{1}{\\left\\{ g(x) \\right\\}^2} \\cdot \\left\\{ f &#8216; (x) g(x) &#8211; f(x) g &#8216; (x) \\right\\} \\\\[4pt]<br \/>\n&#038;= \\frac{f &#8216; (x)g(x) &#8211; f(x) g &#8216; (x)}{\\left\\{ g(x) \\right\\}^2}.<br \/>\n\\end{align}\\]\n<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.2.2. (\uae30\ubcf8 \ubbf8\ubd84 \uacf5\uc2dd)<\/span><\/p>\n<p>\\(k\\)\uac00 \uc0c1\uc218\uc774\uace0, \\(f\\)\uc640 \\(g\\)\uac00 \uac19\uc740 \uc815\uc758\uc5ed\uc744 \uac00\uc9c0\uba70 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\displaystyle\\frac{d}{dx} \\left\\{ kf(x) \\right\\} = kf &#8216; (x) \\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle\\frac{d}{dx} \\left\\{ f(x) + g(x) \\right\\} = f &#8216; (x) + g &#8216; (x)\\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle\\frac{d}{dx} \\left\\{ f(x) &#8211; g(x) \\right\\} = f &#8216; (x) &#8211; g &#8216; (x)\\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle\\frac{d}{dx} \\left\\{ f(x) g(x) \\right\\} = f &#8216; (x) g (x) + f(x) g &#8216; (x)\\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle\\frac{d}{dx} \\left\\{ \\frac{f(x)}{g(x)}\\right\\} = \\frac{f &#8216; (x) g(x) &#8211; f(x) g &#8216; (x)}{\\left\\{ g(x) \\right\\}^2} \\) &nbsp; (\ub2e8, \\(g(x)\\ne 0\\)\uc77c \ub54c.)<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.1.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\(y=x^4\\)\uc774\uba74 \\(y &#8216; = 4x^3\\)\uc774\uace0 \\(y &#8221; = 12x^2\\)\uc774\ub2e4.<\/li>\n<li class=\"margintop1\">\\(f(x)=4x^7\\)\uc774\uba74 \\(f &#8216; (x) = 28x^6\\)\uc774\ub2e4.<\/li>\n<li class=\"margintop1\">\\(y=4x^7 + x^4 &#8211; 3x\\)\uc774\uba74 \\(y &#8216; = 28x^6 + 4x^3 &#8211; 3\\)\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.2.<\/span><\/p>\n<ol class=\"parenthesis\">\n<li>\\(g(x) = (x-1)^2 (x+3)\\)\uc774\uba74<br \/>\n\\[g(x) = (x^2 &#8211; 2x+1)(x+3) = x^3 +x^2 -5x +3\\]<br \/>\n\uc774\ubbc0\ub85c \\(g &#8216; (x) = 3x^2 + 2x-5\\)\uc774\ub2e4.<\/li>\n<li class=\"margintop1\">\\(h(x) = \\displaystyle \\frac{1}{x}\\)\uc774\uba74<br \/>\n\\[h &#8216; (x) = \\frac{-1}{x^2} = &#8211; \\frac{1}{x^2}\\]<br \/>\n\uc774\ub2e4.<\/li>\n<li class=\"margintop1\">\\(k(x) = \\displaystyle\\frac{x+3}{x^2 -4x +5}\\)\uc774\uba74<br \/>\n\\[k &#8216; (x) = \\frac{(x^2 &#8211;  4x+5) &#8211; (x+3)(2x-4)}{(x^2 -4x+5)^2} = \\frac{-x^2 -2x+17}{(x^2 -4x+5)^2}\\]<br \/>\n\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n<p>\ub450 \uc2e4\ud568\uc218 \\(f:A\\rightarrow B\\)\uc640 \\(g:B\\rightarrow C\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0, \\(x_0 \\in A,\\) \\(y_0 = f(x_0 )\\)\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(g\\)\uac00 \\(y_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \\(y_0\\)\uc758 \uadfc\ubc29\uc5d0\uc11c \ud568\uc218 \\(h\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<br \/>\n\\[h(y) = \\begin{cases}<br \/>\n\\displaystyle\\frac{g(y) -g(y_0)}{y-y_0} &#8211; g'(y_ 0) &#038; \\quad \\text{if} \\,\\, y\\ne y_0 ,\\\\[6pt]<br \/>\n0 &#038; \\quad \\text{if} \\,\\, y=y_0 .<br \/>\n\\end{cases}\\]<br \/>\n\uadf8\ub7ec\uba74 \\(h\\)\ub294 \\(y_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4.<\/p>\n<p>\\(y=f(x)\\)\ub77c\uace0 \ud558\uba74<br \/>\n\\[g(f(x)) &#8211; g(f(x_0 )) = g(y) &#8211; g(y_0 ) = g&#8217; (y_0 )(y-y_0 ) + (y-y_0 )h(y)\\]<br \/>\n\uc774\ub2e4. \\(h\\)\uac00 \\(y_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\left. \\frac{d}{dx} (g\\circ f) \\right\\vert_{x=x_0}<br \/>\n&#038;= \\lim_{x\\rightarrow x_0} \\frac{g(f(x)) &#8211; g(f(x_0 ))}{x-x_0} \\\\[5pt]<br \/>\n&#038;= \\lim_{x\\rightarrow x_0} \\left\\{ g &#8216; (f(x_0 )) \\frac{f(x)-f(x_0 )}{x-x_0} + h(f(x))\\frac{f(x)-f(x_0 )}{x-x_0}\\right\\} \\\\[5pt]<br \/>\n&#038;= g &#8216; (f(x_0 )) \\cdot \\lim_{x\\rightarrow x_0} \\frac{f(x)-f(x_0 )}{x-x_0} + h(y_0 ) \\cdot \\lim_{x\\rightarrow x_0} \\frac{f(x) &#8211; f(x_0 )}{x-x_0} \\\\[6pt]<br \/>\n&#038;= g &#8216; ( f(x_0 )) f &#8216; (x_0 ) .<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \\(g\\circ f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \uadf8 \ubbf8\ubd84\uacc4\uc218\ub294<br \/>\n\\[(g\\circ f) &#8216; (x_0 ) = g &#8216; (f(x_0 )) f &#8216; (x_0 )\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.2.3. (\uc5f0\uc1c4 \ubc95\uce59; \ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84\ubc95; The Chain Rule)<\/span><\/p>\n<p>\ub450 \uc2e4\ud568\uc218 \\(f:A\\rightarrow B\\)\uc640 \\(g:B\\rightarrow C\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uace0, \\(x_0 \\in A\\)\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(g\\)\uac00 \\(f(x_0 )\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ud569\uc131\ud568\uc218 \\(g\\circ f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \ubbf8\ubd84\uacc4\uc218\ub294<br \/>\n\\[(g\\circ f) &#8216; (x_0 ) = g &#8216; (f(x_0 )) f &#8216; (x_0 )\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.3.<\/span><br \/>\n\\(f(x)= (2x+3)^10\\)\uc774\uba74<br \/>\n\\[f &#8216; (x) = 10(2x+3)^9 \\times 2 = 20 (2x+3)^9\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.4.<\/span><br \/>\n\uac70\ub4ed\uc81c\uacf1 \ubc95\uce59(\uc815\ub9ac 5.2.1)\uc744 \uc9c0\uc218\uac00 \uc815\uc218\uc778 \uacbd\uc6b0\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\\(m\\)\uc774 \uc74c\uc758 \uc815\uc218\uc774\uace0 \\(f(x)=x^m\\)\uc774\ub77c\uace0 \ud558\uc790. \\(n=-m\\)\uc774\ub77c\uace0 \ud558\uba74 \\(n\\)\uc740 \uc591\uc758 \uc815\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x) &#038;= \\frac{d}{dx}x^m \\\\[4pt]<br \/>\n&#038;= \\frac{d}{dx} \\left\\{ \\frac{1}{x^n}\\right\\} \\\\[4pt]<br \/>\n&#038;= \\frac{-nx^{n-1}}{x^{2n}} \\\\[4pt]<br \/>\n&#038;= \\frac{-n}{x^{n+1}} \\\\[4pt]<br \/>\n&#038;= -nx^{-n-1}\\\\[6pt]<br \/>\n&#038;= mx^{m-1}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uc774 \uacf5\uc2dd\uc740 \\(x\\ne 0\\)\uc77c \ub54c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4\ub294 \uc810\uc744 \uc5fc\ub450\uc5d0 \ub450\uc790.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc5ed\ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n<p>\\(I = [a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc21c\uc99d\uac00\ud558\ub294 \uc2e4\ud568\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c<br \/>\n\\[J = f(I) = \\left\\{ f(x) \\,\\vert\\, x\\in I \\right\\}\\]<br \/>\n\ub77c\uace0 \ud558\uace0, \\(g\\)\uac00 \ud568\uc218 \\(f : I \\rightarrow J\\)\uc758 \uc5ed\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(x_0 (a,\\,b)\\)\uc774\uace0 \\(y_0 = f(x_0 )\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216; (x_0 ) \\ne 0\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790.<\/p>\n<p>\\(\\left\\{ t_n \\right\\}\\)\uc774 \uc138 \uc870\uac74<\/p>\n<ul>\n<li>\\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(t_n \\rightarrow y_0 ,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(t_n \\ne y_0 ,\\)<\/li>\n<li>\uc784\uc758\uc758 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(t_n \\in J\\)<\/li>\n<\/ul>\n<p>\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc784\uc758\uc758 \uc218\uc5f4\uc774\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc77c\ub300\uc77c\ub300\uc751\uc774\ubbc0\ub85c \uac01 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \\(t_n = f(s_n )\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(s_n\\)\uc774 \\(I\\)\uc5d0 \ub531 \ud558\ub098 \uc874\uc7ac\ud55c\ub2e4. \ub354\uc6b1\uc774 \\(g\\)\uac00 \\(y_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(n\\rightarrow\\infty\\)\uc77c \ub54c \\(s_n \\rightarrow x_0\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\ng &#8216; (f(x_0 ))<br \/>\n&#038;= \\lim_{y\\rightarrow y_0} \\frac{g(y) &#8211; g(y_0 )}{y-y_0} \\\\[4pt]<br \/>\n&#038;= \\lim_{n\\rightarrow\\infty} \\frac{g(t_n ) &#8211; g(f(x_0 ))}{t_n &#8211; y_0} \\\\[4pt]<br \/>\n&#038;= \\lim_{n\\rightarrow\\infty} \\frac{s_n &#8211; x_0}{f(s_n ) &#8211; f(x_0 )} \\\\[4pt]<br \/>\n&#038;= \\frac{1}{f &#8216; (x_0 )}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc21c\uac10\uc18c\ud558\ub294 \uacbd\uc6b0\uc5d0\ub3c4 \uac19\uc740 \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 5.2.4. (\uc5ed\ud568\uc218\uc758 \ubbf8\ubd84)<\/span><\/p>\n<p>\\(I = [a,\\,b]\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c, \uc21c\uc99d\uac00\ud558\uac70\ub098 \uc21c\uac10\uc18c\ud558\ub294 \ud568\uc218\uc774\uba70 \\(g\\)\uac00 \\(f\\)\uc758 \uc5ed\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(x_0 \\in (a,\\,b)\\)\uc774\uba70 \\(y_0 = f(x_0 )\\)\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uac00 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216; (x_0 ) \\ne 0\\)\uc774\uba74, \\(g\\)\uac00 \\(y_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \ubbf8\ubd84\uacc4\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[g &#8216; (y_0 ) = \\frac{1}{f &#8216; (x_0 )} .\\]<br \/>\n\uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4.<br \/>\n\\[g &#8216; (f(x_0 )) = \\frac{1}{f &#8216; (x_0 )}.\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.5.<\/span><br \/>\n\uac70\ub4ed\uc81c\uacf1 \ubc95\uce59(\uc815\ub9ac 5.2.1)\uc744 \uc9c0\uc218\uac00 \uc720\ub9ac\uc218\uc778 \uacbd\uc6b0\ub85c \ud655\uc7a5\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\\(f(x)=x^r\\)\uc774\uace0 \\(x > 0\\)\uc774\uba70 \\(r\\)\uac00 \uc591\uc758 \uc720\ub9ac\uc218\ub77c\uace0 \ud558\uc790. \\(r = p\/q\\)\uc774\uba70 \\(p\\)\uc640 \\(q\\)\uac00 \uc591\uc758 \uc815\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f(x)=(x^p )^{1\/q}\\)\uc774\ub2e4.<\/p>\n<p>\\(g(x)=x^q\\)\uc774\uace0 \\(h(y) = y^{1\/q}\\)\uc774\uba70 \\(x > 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(x > 0\\)\uc778 \ubc94\uc704\uc5d0\uc11c \\(g\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \uc21c\uc99d\uac00\ud558\ubbc0\ub85c, \\(g\\)\uc758 \uc5ed\ud568\uc218 \\(h(y)\\)\uac00 \\(y > 0\\)\uc778 \ubc94\uc704\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \uadf8\ub7f0\ub370 \\(f(x) = h(x^p )\\)\uc774\ubbc0\ub85c \\(f\\) \ub610\ud55c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\\(f(x)=(x^p )^{1\/q}\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\left\\{ f(x) \\right\\}^q = x^{qr} = x^p\\]<br \/>\n\uc774\ub2e4. \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uba74<br \/>\n\\[q \\left\\{ f(x) \\right\\}^{q-1} f &#8216; (x) = px^{p-1}\\]<br \/>\n\uc774\uba70, \uc774 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[f &#8216; (x) = \\frac{p}{q} \\cdot \\frac{x^{p-1}}{\\left\\{ f(x) \\right\\} ^{q-1}}.\\]<br \/>\n\\(x^{p\/q}\\)\uc744 \\(x^r\\)\uc73c\ub85c \ubc14\uafb8\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x)<br \/>\n&#038;= \\frac{p}{q}\\cdot \\frac{x^{p-1}}{\\left\\{ x^{p\/q}\\right\\}^{q-1}}\\\\[4pt]<br \/>\n&#038;= \\frac{p}{q}\\cdot \\frac{x^{p-1}}{x^{p(q-1)\/q}} \\\\[4pt]<br \/>\n&#038;= \\frac{p}{q}\\cdot x^{(p\/q)-1} \\\\[6pt]<br \/>\n&#038;= rx^{r-1}.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc74c\ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n<p>\\(y\\)\uac00 \\(x\\)\uc758 \ud568\uc218\uc774\uace0 \ub450 \ubcc0\uc218\uc758 \uad00\uacc4\uac00 \\(F(x,\\,y)=0\\) \uaf34\uc758 \uc2dd\uc73c\ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(y\\)\ub97c <span class=\"defined\">\uc74c\uc801\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218<\/span>(implicitly-defined function) \ub610\ub294 \uac04\ub2e8\ud788 <span class=\"defined\">\uc74c\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \uc74c\uc801\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.6.<\/span><br \/>\n\\(y\\)\uac00 \\(x\\)\uc758 \ud568\uc218\uc774\uace0<br \/>\n\\[x^2 + 4xy^5 + 7xy + 8 =0\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790. \ub610\ud55c \\(y\\)\uac00 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc8fc\uc5b4\uc9c4 \uc2dd\uc758 \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[2x + 4y^5 + 20xy^4 \\frac{dy}{dx} + 7y + 7x \\frac{dy}{dx}= 0.\\]<br \/>\n\uc774 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[2x + 4y^5 + 7y + (20xy^4 + 7x) \\frac{dy}{dx} = 0.\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\frac{dy}{dx} = &#8211; \\frac{2x+4y^5 + 7y}{20xy^4 + 7x}.\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.7.<\/span><br \/>\n\ud0c0\uc6d0<br \/>\n\\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]<br \/>\n\uc704\uc758 \uc810 \\((x_0 ,\\, y_0 )\\)\uc5d0\uc11c \ud0c0\uc6d0\uc5d0 \uc811\ud558\ub294 \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud574 \ubcf4\uc790. \uc704 \ub4f1\uc2dd\uc758 \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uba74<br \/>\n\\[\\frac{2x}{a^2} + \\frac{2y}{b^2} \\cdot \\frac{dy}{dx} = 0\\]<br \/>\n\uc774\ub2e4. \ub9cc\uc57d \\(y\\ne 0\\)\uc774\uba74<br \/>\n\\[\\frac{dy}{dx} = &#8211; \\frac{b^2 x}{a^2 y}\\]<br \/>\n\uc774\ub2e4. \\(x_0\\)\uc640 \\(y_0\\)\ub97c \uac01\uac01 \\(x\\)\uc640 \\(y\\)\uc5d0 \ub300\uc785\ud558\uba74 \uc811\uc120\uc758 \uae30\uc6b8\uae30<br \/>\n\\[ &#8211; \\frac{b^2 x_0}{a^2 y_0}\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[y = &#8211; \\frac{b^2 x_0}{a^2 y_0} (x-x_0 ) + y_0 = &#8211; \\frac{b^2 x_0}{a^2 y_0} x + \\frac{b^2 x_0 ^2}{a^2 y_0} + y_0 .\\]<br \/>\n\uc774 \uc2dd\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\frac{x_0 x}{a^2} + \\frac{y_0 y}{b^2} = \\frac{x_0 ^2}{a^2} + \\frac{y_0 ^2}{b^2} .\\]<br \/>\n\uadf8\ub7f0\ub370 \\((x_0 ,\\, y_0 )\\)\uc774 \ud0c0\uc6d0 \uc704\uc758 \uc810\uc774\ubbc0\ub85c<br \/>\n\\[\\frac{x_0 ^2}{a^2} + \\frac{y_0 ^2}{b^2} = 1\\]<br \/>\n\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \ub530\ub77c\uc11c \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc740<br \/>\n\\[\\frac{x_0 x}{a^2} + \\frac{y_0 y}{b^2} = 1\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8 \\(y=0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(x = \\pm a\\)\uc774\uba70, \uc774\ub54c \uc811\uc120\uc740 \\(y\\)\ucd95\uacfc \ud3c9\ud589\ud558\ub2e4. \uc989 \uc774 \uacbd\uc6b0 \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc740<br \/>\n\\[x=\\pm a\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\((x_0 ,\\, y_0 ) = (\\pm a ,\\, 0)\\)\uc774\ubbc0\ub85c, \uc774 \uacbd\uc6b0\uc5d0\ub3c4 \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc744<br \/>\n\\[\\frac{x_0 x}{a^2} + \\frac{y_0 y}{b^2} = 1\\]<br \/>\n\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ub9e4\uac1c\ubcc0\uc218\ub85c \ub098\ud0c0\ub09c \ud568\uc218\uc758 \ubbf8\ubd84<\/h2>\n<p>\\(I\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(t\\)\uac00 \\(I\\)\uc758 \uac12\uc744 \ucde8\ud558\ub294 \ubcc0\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(y\\)\uac00 \\(x\\)\uc758 \ud568\uc218\uc774\uace0 \\(x = f(t),\\) \\(y=g(t)\\)\uac00 \uc131\ub9bd\ud55c\ub2e4\uace0 \ud558\uc790. \ub9cc\uc57d \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc774\uace0 \\(f &#8216; (t) \\ne 0\\)\uc774\uba74<br \/>\n\\[\\frac{dy}{dx} = \\frac{\\displaystyle\\frac{dy}{dt}}{\\displaystyle\\frac{dx}{dt}} = \\frac{g &#8216; (t)}{f &#8216; (t)}\\]<br \/>\n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.2.8.<\/span><br \/>\n\\(y\\)\uac00 \\(x\\)\uc758 \ud568\uc218\uc774\uace0 \\(x=2t+1,\\) \\(y=-4t^2 -3\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[\\frac{dx}{dt} = 2 ,\\quad \\frac{dy}{dt} = -8t\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\frac{dy}{dx} = \\frac{\\displaystyle\\frac{dy}{dt}}{\\displaystyle\\frac{dx}{dt}} = \\frac{-8t}{2} = -4t\\]<br \/>\n\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/definition-of-a-derivative\">\ubbf8\ubd84\uc758 \uc815\uc758<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/describing-graphs\">\ud568\uc218\uc758 \uadf8\ub798\ud504<\/a><\/li>\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 5\uc7a5 2\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uae30\ubcf8 \uacf5\uc2dd \\(n\\)\uc774 \uc591\uc758 \uc815\uc218\uc774\uace0 \\(x\\)\uac00 \uc2e4\ubcc0\uc218\uc774\uba70 \\(f(x) = x^n\\)\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \ub9cc\uc57d \\(n=1\\)\uc774\uba74 \\(f &#8216; (x) = \\lim_{\\Delta x \\rightarrow 0} \\frac{f(x+\\Delta x) &#8211; f(x)}{\\Delta x} = \\lim_{\\Delta x \\rightarrow 0} \\frac{(x+\\Delta x) -x}{\\Delta x} = 1\\) \uc774\ub2e4. \ub9cc\uc57d \\(n \\ge 2\\)\uc774\uba74 \\(\\begin{align} f &#8216; (x) &#038;= \\lim_{\\Delta&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":502,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6714","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6714","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=6714"}],"version-history":[{"count":35,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6714\/revisions"}],"predecessor-version":[{"id":7997,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6714\/revisions\/7997"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6714"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}