{"id":1948,"date":"2019-03-18T12:01:25","date_gmt":"2019-03-18T03:01:25","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1948"},"modified":"2019-09-05T19:51:48","modified_gmt":"2019-09-05T10:51:48","slug":"calculus-the-differentiation-rules","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-the-differentiation-rules\/","title":{"rendered":"\uc5ec\ub7ec \uac00\uc9c0 \ubbf8\ubd84 \ubc95\uce59"},"content":{"rendered":"<p>\uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \ubbf8\ubd84 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uba74 \uc720\ub9ac\ud568\uc218\uc758 \ubbf8\ubd84\uc740 \ud560 \uc218 \uc788\uc9c0\ub9cc, \uadf8 \uc678\uc758 \ubcf5\uc7a1\ud55c \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uae30\uc5d0\ub294 \uc5b4\ub824\uc6c0\uc774 \uc788\ub2e4. \ud569\uc131\ud568\uc218, \uc74c\ud568\uc218, \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \ub354 \ub2e4\uc591\ud55c \uc885\ub958\uc758 \ud568\uc218\ub97c \ubbf8\ubd84\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p><!-- #### #### #### #### #### #### #### #### #### #### #### #### --><\/p>\n<h3>\ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n<p>\ub2e4\uc74c\uacfc \uac19\uc740 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790.<br \/>\n\\[h(x) = (2x+4)^3 \\tag{1}\\]<br \/>\n\uc774 \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uace0 \ubbf8\ubd84\uacc4\uc218 \\(h &#8216; (1)\\)\uc744 \uad6c\ud574 \ubcf4\uc790. \uc6b0\ubcc0\uc744 \uc804\uac1c\ud558\uba74<br \/>\n\\[h(x) = 8x^3 + 48x^2 + 96x + 64 \\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[h &#8216; (x) = 24x^2 + 96x + 96\\]<br \/>\n\uc774\uace0<br \/>\n\\[h &#8216; (1) = 24 + 96 + 96 = 216\\]<br \/>\n\uc774\ub2e4. \ud558\uc9c0\ub9cc (1)\uc758 \uc6b0\ubcc0\uc744 \uc804\uac1c\ud558\uc9c0 \uc54a\uace0 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \\(h(x)\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[u = f(x) = 2x+4 ,\\quad y = g(u) = u^3\\]<br \/>\n\uc774\ub77c\uace0 \ud558\uba74<br \/>\n\\[\\frac{du}{dx} = f &#8216; (x) = 2 ,\\quad \\frac{dy}{du} = g &#8216; (u) = 3u^2\\]<br \/>\n\uc774\ub2e4. \\(x=1\\)\uc758 \uadfc\ucc98\uc5d0\uc11c \\(u=f(x)\\)\uc758 \ubcc0\ud654\ub7c9\uc740 \\(x\\)\uc758 \ubcc0\ud654\ub7c9\uc758 \\(2\\)\ubc30\uc774\uace0, \\(u = 6 = f(1)\\)\uc758 \uadfc\ucc98\uc5d0\uc11c \\(y=g(u)\\)\uc758 \ubcc0\ud654\ub7c9\uc740 \\(u\\)\uc758 \ubcc0\ud654\ub7c9\uc758 \\(3 \\times 6^2 = 108\\)\ubc30\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x=1\\) \uadfc\ucc98\uc5d0\uc11c \\(y=g(f(x))\\)\uc758 \ubcc0\ud654\ub7c9\uc740 \\(x\\)\uc758 \ubcc0\ud654\ub7c9\uc758 \\(2 \\times 108 = 216\\)\ubc30\uc774\ub2e4. \uc774\uac83\uc744 \uc2dd\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74<br \/>\n\\[ \\left. \\frac{dy}{dx} \\right\\vert _{x=1} = \\left. \\frac{du}{dx}\\right\\vert_{x=1} \\cdot \\left. \\frac{dy}{du} \\right\\vert_{u=f(1)}\\]<br \/>\n\uc989<br \/>\n\\[h &#8216; (1) = f &#8216; (1) \\cdot g &#8216; (f(1))\\]<br \/>\n\uc774\ub2e4. \uc774\uc640 \uac19\uc774 \\(h\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ub450 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uc758 \ud569\uc131 \\(h(x) = g(f(x))\\)\ub85c \uc815\uc758\ub418\uc5b4 \uc788\uc744 \ub54c, \\(h\\)\uc758 \ub3c4\ud568\uc218\ub294<br \/>\n\\[h &#8216; (x) = g &#8216; (f(x)) f &#8216; (x)\\]<br \/>\n\uac00 \ub418\ub294\ub370, \uc774\ub7ec\ud55c \ubbf8\ubd84 \ubc95\uce59\uc744 <span class=\"defined\">\uc5f0\uc1c4 \ubc95\uce59<\/span>(chain rule)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 1. (\uc5f0\uc1c4 \ubc95\uce59)<\/span><\/p>\n<p>\ud568\uc218 \\(u=f(x)\\)\uac00 \uc810 \\(x_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \ud568\uc218 \\(y = g(u)\\)\uac00 \uc810 \\(f(x_0 )\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[y = h(x) = g(f(x))\\]<br \/>\n\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(h\\)\ub294 \uc810 \\(x_0 \\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0<br \/>\n\\[ (g \\circ f) &#8216; (x_0 ) = g &#8216; (f(x_0 )) f &#8216; (x_0 )\\tag{2}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub77c\uc774\ud504\ub2c8\uce20\uc758 \ud45c\uae30\ubc95\uc744 \uc774\uc6a9\ud558\uba74 (2)\ub294<br \/>\n\\[\\left.\\frac{dy}{dx}\\right\\vert_{x=x_0} = \\left.\\frac{dy}{du}\\right\\vert_{u=f(x_0 )} \\cdot \\left.\\frac{du}{dx}\\right\\vert_{x=x_0}\\]<br \/>\n\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p>\\(u_0 = f(x_0 )\\)\uc774\ub77c\uace0 \ud558\uc790. \\(x\\)\uc758 \uc99d\ubd84\uc744 \\(\\Delta x\\)\ub77c\uace0 \ud558\uace0, \uc774 \uc99d\ubd84\uc5d0 \ub530\ub978 \\(u\\)\uc640 \\(y\\)\uc758 \uc99d\ubd84\uc744 \uac01\uac01 \\(\\Delta u ,\\) \\(\\Delta y\\)\ub77c\uace0 \ud558\uc790. \\(f\\)\ub294 \\(x_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(g\\)\ub294 \\(u_0 = f(x_0 )\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(\\Delta x \\to 0\\)\uc77c \ub54c \\(\\Delta u \\to 0 ,\\) \\(\\Delta y \\to 0\\)\uc774\ub2e4.<\/p>\n<p>\ubbf8\ubd84\uc18c\uc640 \uad00\ub828\ub41c \ud568\uc22b\uac12\uc758 \uc99d\ubd84 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uba74 \\(u\\)\uc758 \uc99d\ubd84 \\(\\Delta u\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\lim_{\\Delta x\\to 0} \\epsilon_1 = 0\\]<br \/>\n\uc778 \ud568\uc218 \\(\\epsilon_1\\)\uc774 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[\\Delta u = f &#8216; (x_0 ) \\Delta x + \\epsilon_1 \\Delta x = (f &#8216; (x_0) + \\epsilon_1 )\\Delta x\\tag{3}\\]<br \/>\n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \\(y\\)\uc758 \uc99d\ubd84 \\(\\Delta y\\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\lim_{\\Delta u\\to 0} \\epsilon_2 =0\\]<br \/>\n\uc778 \ud568\uc218 \\(\\epsilon_2\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[\\Delta y = g &#8216; (u_0 ) \\Delta u + \\epsilon_2 \\Delta u = (g &#8216; (u_0) + \\epsilon_2 )\\Delta u\\tag{4}\\]<br \/>\n\ub85c \uc4f8 \uc218 \uc788\ub2e4. \ub450 \uc2dd (3)\uacfc (4)\ub97c \uacb0\ud569\ud558\uba74<br \/>\n\\[\\Delta y = (g &#8216; (u_0 ) + \\epsilon_2 )(f &#8216; (x_0 ) + \\epsilon_1 ) \\Delta x\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\n\\left.\\frac{dy}{dx}\\right\\vert_{x=x_0}<br \/>\n&#038;= \\lim_{\\Delta x \\to 0} \\frac{\\Delta y}{\\Delta x} \\\\[6pt]<br \/>\n&#038;= \\lim_{\\Delta x \\to 0} \\frac{(g &#8216; (u_0 ) + \\epsilon_2 )(f &#8216; (x_0 ) + \\epsilon_1 ) \\Delta x}{\\Delta x} \\\\[6pt]<br \/>\n&#038;= \\lim_{\\Delta x \\to 0} (g &#8216; (u_0 ) + \\epsilon_2 ) \\cdot \\lim_{\\Delta x \\to 0} (f &#8216; (x_0 ) + \\epsilon_1 ) \\\\[8pt]<br \/>\n&#038;= \\lim_{\\Delta u \\to 0} (g &#8216; (u_0 ) + \\epsilon_2 ) \\cdot \\lim_{\\Delta x \\to 0} (f &#8216; (x_0 ) + \\epsilon_1 ) \\\\[8pt]<br \/>\n&#038;= g &#8216; (u_0 ) \\cdot f &#8216; (x_0 ) \\\\[8pt]<br \/>\n&#038;= g &#8216; (f(x_0 )) \\cdot f &#8216; (x_0 )<br \/>\n\\end{align}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofbegin\"><span class=\"proof\">\ub2e4\ub978 \ubc29\ubc95\uc758 \uc99d\uba85.<\/span><br \/>\n\\(u_0 = f(x_0 )\\)\uc774\ub77c\uace0 \ud558\uc790. \ud568\uc218 \\(g\\)\uac00 \\(u_0\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \ud568\uc218 \\(\\eta\\)\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<br \/>\n\\[\\eta (u) = \\begin{cases}<br \/>\n\\frac{g(u)-g(u_0 )}{u-u_0} &#8211; g &#8216;(u_0 ) &#038;\\quad \\text{if}\\,\\, u \\ne u_0 \\\\[6pt]<br \/>\n0 &#038;\\quad \\text{if}\\,\\, u = u_0<br \/>\n\\end{cases}\\]<br \/>\n\\(u=f(x)\\)\ub77c\uace0 \ud558\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n(g\\circ f)(x) &#8211; (g \\circ f)(x_0 )<br \/>\n&#038;= g(f(x)) &#8211; g(f(x_0 )) \\\\[8pt]<br \/>\n&#038;= g(u) &#8211; g(u_0 ) \\\\[8pt]<br \/>\n&#038;= g &#8216; (u_0 )(u-u_0 ) + (u-u_0 )\\eta(u)<br \/>\n\\end{align}\\]<br \/>\n\uadf8\ub7f0\ub370 \\(\\eta\\)\ub294 \\(u_0\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n(g\\circ f) &#8216; (x_0 )<br \/>\n&#038;= \\lim_{x\\to x_0} \\frac{g(f(x)) &#8211; g(f(x_0 ))}{x-x_0} \\\\[6pt]<br \/>\n&#038;= \\lim_{x\\to x_0}\\left[g &#8216; (f(x_0 )) \\frac{f(x) &#8211; f(x_0 )}{x-x_0 } + \\eta (f(x))\\frac{f(x)-f(x_0 )}{x-x_0 }\\right]\\\\[6pt]<br \/>\n&#038;= g &#8216; (f(x_0 )) \\lim_{x\\to x_0} \\frac{f(x)-f(x_0 )}{x-x_0} + \\eta(u_0 ) \\lim_{x\\to x_0} \\frac{f(x)-f(x_0 )}{x-x_0}\\\\[6pt]<br \/>\n&#038;= g &#8216; (f(x_0 )) f &#8216; (x_0 ). \\tag*{\\(\\blacksquare\\)}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span><br \/>\n\ub2e4\uc74c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[h(x) = \\sin (2x^3 +4 ).\\]<br \/>\n\\(f(x) = 2x^3 +4,\\) \\(g(u) = \\sin u\\)\ub77c\uace0 \ud558\uba74 \\(h(x) = g(f(x))\\)\uc774\ubbc0\ub85c<br \/>\n\\[\\begin{align}<br \/>\nh &#8216; (x)<br \/>\n&#038;= g &#8216; (f(x)) \\cdot f &#8216; (x)  \\\\[8pt]<br \/>\n&#038;= \\cos (f(x)) \\cdot (6x^2 ) \\\\[8pt]<br \/>\n&#038;= \\cos(2x^3 +4) \\cdot (6x^2 ). \\tag*{\\(\\square\\)}<br \/>\n\\end{align}\\]\n<\/div>\n<p>\uacf5\uc2dd (2)\ub294 (\ubc14\uae65 \ubbf8\ubd84)\u00d7(\uc18d \ubbf8\ubd84)\uc774\ub77c\uace0 \uae30\uc5b5\ud558\uba74 \ud3b8\ub9ac\ud558\ub2e4. \ud2b9\ud788 \uc138 \uac1c \uc774\uc0c1\uc758 \ud568\uc218\ub97c \ud569\uc131\ud558\uc5ec \ub9cc\ub4e0 \ud568\uc218\ub97c \ubbf8\ubd84\ud560 \ub54c\uc5d0\ub3c4 \uc774 \uacf5\uc2dd\uc744 \uadf8\ub300\ub85c \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[h(x) = k(g(f(x)))\\]<br \/>\n\uc77c \ub54c<br \/>\n\\[h &#8216; (x) = k &#8216; (g(f(x))) \\cdot g &#8216; (f(x)) \\cdot f &#8216; (x)\\tag{5}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 2.<\/span><br \/>\n\ub2e4\uc74c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[f(x) = \\cos^3 ( 4x^2 -x+7).\\]<br \/>\n\uc774 \ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc138 \uac1c\uc758 \ud568\uc218<br \/>\n\\[\\begin{align}<br \/>\ny &#038;= u^3 ,\\\\[8pt]<br \/>\nu &#038;= \\cos t ,\\\\[8pt]<br \/>\nt &#038;= 4x^2 -x+7<br \/>\n\\end{align}\\]<br \/>\n\uc774 \ud569\uc131\ub41c \uac83\uc73c\ub85c \ubcfc \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5f0\uc1c4 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\nf &#8216; (x)<br \/>\n&#038;= \\frac{dy}{dx} \\\\[6pt]<br \/>\n&#038;= \\frac{dy}{du} \\cdot \\frac{du}{dt} \\cdot \\frac{dt}{dx} \\\\[6pt]<br \/>\n&#038;= 3u^2 \\cdot (-\\sin t) \\cdot (8x-1) \\\\[8pt]<br \/>\n&#038;= &#8211; 3 \\cos^2 (4x^2 -x+7) \\cdot \\sin(4x^2 -x+7) \\cdot (8x-1).\\tag*{\\(\\square\\)}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- #### #### #### #### #### #### #### #### #### #### #### #### --><\/p>\n<h3>\uc74c\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n<p>\ud568\uc218 \\(y=f(x)\\)\uc758 \ub450 \ubcc0\uc218 \\(x\\)\uc640 \\(y\\)\uac00 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4\uace0 \ud558\uc790.<br \/>\n\\[y^2 =x.\\tag{6}\\]<br \/>\n\uc774\ub54c \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574\ubcf4\uc790. (6)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 \\(f(x) = \\sqrt{x}\\)\uc77c \uc218\ub3c4 \uc788\uace0 \\(f(x) = -\\sqrt{x}\\)\uc77c \uc218\ub3c4 \uc788\ub2e4. \ub450 \ud568\uc218\ub97c \uad6c\ubd84\ud558\uae30 \uc704\ud558\uc5ec<br \/>\n\\[\\begin{align}<br \/>\nf_1 (x) &#038;= \\sqrt{x}, \\\\[8pt]<br \/>\nf_2 (x) &#038;= &#8211; \\sqrt{x}<br \/>\n\\end{align}\\]<br \/>\n\ub77c\uace0 \ud558\uc790. \uc774 \ub450 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub294<br \/>\n\\[\\begin{align}<br \/>\nf_1 &#8216; (x) &#038;= \\frac{1}{2\\sqrt{x}}, \\\\[6pt]<br \/>\nf_2 &#8216; (x) &#038;= -\\frac{1}{2\\sqrt{x}}<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \uc774 \ub450 \ub3c4\ud568\uc218\ub97c \uad6c\ud55c \uac83\ub9cc\uc73c\ub85c\ub294 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud588\ub2e4\uace0 \ud560 \uc218 \uc5c6\ub2e4. \uc65c\ub0d0\ud558\uba74 (6)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218\ub294 \\(f_1\\)\uacfc \\(f_2\\) \uc678\uc5d0\ub3c4 \ubb34\uc218\ud788 \ub9ce\uae30 \ub54c\ubb38\uc774\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[f_3 (x) = \\begin{cases}<br \/>\n\\sqrt{x} \\quad&#038;\\text{if}\\,\\, x > \\pi ,\\\\[8pt]<br \/>\n-\\sqrt{x} \\quad&#038;\\text{if}\\,\\, 0 \\le x \\le \\pi<br \/>\n\\end{cases}\\]<br \/>\n\ub294 (6)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ub610\ub2e4\ub978 \ud568\uc218\uc774\ub2e4.<\/p>\n<p>\ub300\uc2e0 \ub2e4\ub978 \ubc29\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. (6)\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \uc2dd<br \/>\n\\[y^2 = x\\]<br \/>\n\uc758 \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud55c\ub2e4. \uc5ec\uae30\uc11c \uc88c\ubcc0\uc740 \uc5f0\uc1c4 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84\ud55c\ub2e4. \uadf8\ub7ec\uba74<br \/>\n\\[2y \\frac{dy}{dx} = 1\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\frac{dy}{dx} = \\frac{1}{2y} = \\frac{1}{2f(x)} \\tag{7}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \ube44\ub85d \uc774 \uc2dd\uc740 \\(x\\)\uc5d0 \ub300\ud55c \uc2dd\uc774 \uc544\ub2c8\uc9c0\ub9cc, (6)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(f\\)\uac00 \uc8fc\uc5b4\uc9c8 \ub54c\ub9c8\ub2e4 (7)\uc5d0 \ub300\uc785\ud558\uc5ec \\(f\\)\uc5d0 \ub300\uc751\ub418\ub294 \ub3c4\ud568\uc218\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (7)\uc740 (6)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud55c \uac83\uc774\ub77c \ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc774\ucc98\ub7fc \ud568\uc218 \\(f\\)\uac00 \\(y=f(x)\\)\uc758 \uaf34\ub85c \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \u2018<span class=\"defined\">\\(f\\)\ub294 \uc591\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5c8\ub2e4<\/span>\u2019\ub77c\uace0 \ud558\uace0, \\(y=f(x)\\)\uc758 \uaf34\ub85c \uc815\uc758\ub41c \ud568\uc218\ub97c <span class=\"defined\">\uc591\ud568\uc218<\/span>(explicit function)\ub77c\uace0 \ubd80\ub978\ub2e4. \ubc18\uba74\uc5d0 (6)\ucc98\ub7fc \ud568\uc218 \\(f\\)\uac00 \\(F(x,\\,y)=0\\) \ub610\ub294 \uc774\uc640 \ub3d9\uce58\uc778 \uaf34\ub85c \uc815\uc758\ub418\uc5c8\uc744 \ub54c \u2018<span class=\"defined\">\\(f\\)\ub294 \uc74c\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5c8\ub2e4<\/span>\u2019\ub77c\uace0 \ud558\uace0, \uc774\uc640 \uac19\uc774 \uc815\uc758\ub41c \ud568\uc218\ub97c <span class=\"defined\">\uc74c\ud568\uc218<\/span>(implicit function)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc591\ud568\uc218\ub97c <span class=\"defined\">\ub4dc\ub7ec\ub09c \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud558\uace0, \uc74c\ud568\uc218\ub97c <span class=\"defined\">\uc228\uc740 \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc74c\ud568\uc218 \ubbf8\ubd84\ubc95<\/span><\/p>\n<p>\ud568\uc218 \\(y=f(x)\\)\uac00 \\(F(x,\\,y)=0\\)\uacfc \uac19\uc774 \uc74c\uc801\uc73c\ub85c \uc815\uc758\ub418\uc5c8\uc744 \ub54c, \\(f\\)\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(F(x,\\,y)=0\\)\uc758 \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud55c\ub2e4. \uc774\ub54c \\(y\\)\ub97c \\(x\\)\uc758 \ud568\uc218\ub85c \ubcf4\uace0 \uc5f0\uc1c4 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84\ud55c\ub2e4.<\/li>\n<li>\ubbf8\ubd84\ud55c \uc2dd\uc744 \\(dy\/dx\\)\uc5d0 \ub300\ud558\uc5ec \ud47c\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<p>\ub3c4\ud615\uc758 \ubc29\uc815\uc2dd\uc744 \uacf5\ubd80\ud558\uba70 \ubc30\uc6e0\ub358 \uc6d0\uc758 \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc740 \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uad6c\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 3.<\/span><br \/>\n\\(r > 0\\)\uc77c \ub54c, \ubc29\uc815\uc2dd<br \/>\n\\[x^2 + y^2 = r^2\\tag{8}\\]<br \/>\n\uc73c\ub85c \ud45c\ud604\ub418\ub294 \uc6d0 \uc704\uc758 \uc810 \\((x_1 ,\\,y_1)\\)\uc5d0\uc11c \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud574\ubcf4\uc790.<\/p>\n<p>\uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec (8)\uc744 \ubbf8\ubd84\ud558\uba74<br \/>\n\\[2x + 2y \\frac{dy}{dx} =0\\]<br \/>\n\uc774\ubbc0\ub85c \\(y_1 \\ne 0\\)\uc77c \ub54c<br \/>\n\\[\\left.\\frac{dy}{dx}\\right\\vert_{(x,\\,y)=(x_1 ,\\, y_1 )} = -\\frac{x_1}{y_1}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc6d0 \uc704\uc758 \uc810 \\((x_1 ,\\,y_1)\\)\uc5d0\uc11c \uc774 \uc6d0\uc5d0 \uc811\ud558\ub294 \uc9c1\uc120\uc758 \ubc29\uc815\uc2dd\uc740<br \/>\n\\[y=-\\frac{x_1}{y_1}(x-x_1 ) + y_1\\]<br \/>\n\uc989<br \/>\n\\[xx_1 + yy_1  =  x_1 ^2 +y_1 ^2 \\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc5d0 \\(x_1^2 + y_1^2 = r^2\\)\uc744 \ub300\uc785\ud558\uba74 \uad6c\ud558\ub294 \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc740<br \/>\n\\[xx_1 + yy_1 = r^2\\tag{9}\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8 \\(y=0\\)\uc77c \ub54c\uc5d0\ub294 \\(x\\)\ub97c \\(y\\)\uc758 \ud568\uc218\ub85c \ubcf4\uace0 \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec (8)\uc744 \ubbf8\ubd84\ud558\uba74<br \/>\n\\[2x\\frac{dx}{dy} + 2y =0\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\left.\\frac{dx}{dy}\\right\\vert_{(x,\\,y)=(x_1,\\,y_1)} = -\\frac{y_1}{x_1}\\]<br \/>\n\uc744 \uc5bb\uc73c\uba70, \uc774 \ubbf8\ubd84\uacc4\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc811\uc120\uc758 \ubc29\uc815\uc2dd\uc744 \uad6c\ud558\uba74 \uc55e\uc5d0\uc11c\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c (9)\ub97c \uc5bb\ub294\ub2e4.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p>\uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc774\uacc4\ub3c4\ud568\uc218\ub97c \uad6c\ud560 \uc218\ub3c4 \uc788\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 4.<\/span><br \/>\n\\(x\\)\uc640 \\(y\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc740 \uad00\uacc4\ub97c \uac00\uc9c0\uace0 \uc788\uc744 \ub54c \uc774\uacc4\ub3c4\ud568\uc218 \\(d^2 y \/ dx^2\\)\ub97c \uad6c\ud574 \ubcf4\uc790.<br \/>\n\\[2x^3 &#8211; 3y^2 = 8\\]<br \/>\n\uba3c\uc800 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\frac{d}{dx} (2x^3 &#8211; 3y^2 ) &#038;= \\frac{d}{dx}(8), \\\\[6pt]<br \/>\n6x^2 &#8211; 6yy &#8216; &#038;= 0, \\\\[6pt]<br \/>\ny &#8216; &#038;= \\frac{x^2}{y} . \\quad(\\text{when}\\,\\, y \\ne 0)<br \/>\n\\end{align}\\]<br \/>\n\uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc88 \ub354 \ubbf8\ubd84\ud558\uc5ec \uc774\uacc4\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74<br \/>\n\\[y &#8216; &#8216; = \\frac{d}{dx} \\left( \\frac{x^2}{y} \\right) = \\frac{2xy &#8211; x^2 y &#8216; }{y^2} = \\frac{2x}{y} &#8211; \\frac{x^2}{y^2} y \\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(y &#8216; = x^2 \/ y\\)\uc774\ubbc0\ub85c \uc704 \uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc4f8 \uc218 \uc788\ub2e4.<br \/>\n\\[y &#8216; &#8216; = \\frac{2x}{y} &#8211; \\frac{x^2}{y^2} \\left( \\frac{x^2}{y}\\right) = \\frac{2x}{y} &#8211; \\frac{x^4}{y^3}. \\quad (\\text{when}\\,\\,y\\ne 0)\\]\n<\/p>\n<p>(Thomas&#8217; Calculus International Edition 13\ud310 3.7\uc808\uc5d0\uc11c \ubc1c\ucdcc\ud568.)<span class=\"qee\"><\/span><\/p>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\uc8fc\uc758.<\/span><br \/>\n\uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc740 \uc74c\uc801\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218\uac00 <span class=\"defined\">\ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\ub294 \uac00\uc815 \ud558\uc5d0<\/span> \ub3c4\ud568\uc218\ub97c \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc744 \uc81c\uacf5\ud55c\ub2e4. \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \\(f\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud588\ub2e4 \ud558\ub354\ub77c\ub3c4 \\(f\\)\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc740 \ubcf4\uc7a5\ud560 \uc218 \uc5c6\ub2e4. \uc608\ucee8\ub300<br \/>\n\\[f(x) = \\begin{cases}<br \/>\n\\sqrt{x} &#038;\\quad\\text{if} \\,\\, x\\in [0,\\,\\infty )\\cap \\mathbb{Q} \\\\[8pt]<br \/>\n-\\sqrt{x} &#038;\\quad\\text{if} \\,\\, x\\in [0,\\,\\infty )\\setminus \\mathbb{Q}<br \/>\n\\end{cases}\\]<br \/>\n\ub77c\uace0 \ud558\uba74 \\(y=f(x)\\)\uc758 \ubcc0\uc218 \\(x,\\) \\(y\\)\ub294<br \/>\n\\[y^2 = x\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \ub3c4\ud568\uc218\ub97c \uad6c\ud560 \uc218 \uc788\uc744 \uac83\ucc98\ub7fc \ubcf4\uc774\uc9c0\ub9cc \\(f\\)\ub294 \uc5b4\ub290 \uacf3\uc5d0\uc11c\ub3c4 \ubbf8\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4.<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p><!-- #### #### #### #### #### #### #### #### #### #### #### #### --><\/p>\n<h3>\uc5ed\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n<p>\ud568\uc218 \\(y=\\sqrt{x}\\)\uc758 \ub3c4\ud568\uc218\ub294<br \/>\n\\[y &#8216; = \\frac{1}{2\\sqrt{x}}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(y=\\sqrt{x}\\)\ub294 \\(y=f(x)=x^2\\,\\, (x\\ge 0)\\)\uc758 \uc5ed\ud568\uc218\uc774\ub2e4.<br \/>\n\\((a ,\\,b)\\)\uac00 \ud568\uc218 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504 \uc704\uc758 \uc810\uc774\uba74 \\((b ,\\,a)\\)\ub294 \ud568\uc218 \\(y=f^{-1}(x)=\\sqrt{x}\\)\uc758 \uadf8\ub798\ud504 \uc704\uc758 \uc810\uc774\ub2e4. \\(f &#8216; (x) = 2x\\)\uc774\ubbc0\ub85c \\(x=a\\)\uc77c \ub54c \\(y=x^2\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub294 \\(f &#8216; (a ) = 2a\\)\uc774\uba70, \ud568\uc218\uc758 \uadf8\ub798\ud504\uc640 \uc5ed\ud568\uc218\uc758 \uadf8\ub798\ud504\ub294 \uc9c1\uc120 \\(y=x\\)\uc5d0 \ub300\ud558\uc5ec \ub300\uce6d\uc774\ubbc0\ub85c, \\(x=b\\)\uc5d0\uc11c \\(y= f^{-1} (x) = \\sqrt{x}\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub294<br \/>\n\\[\\frac{1}{f &#8216; (a )} = \\frac{1}{2a} = \\frac{1}{2\\sqrt{b}}\\]<br \/>\n\uc774 \ub41c\ub2e4. \uc774\ub85c\uc368<br \/>\n\\[(f^{-1}) &#8216; (b) = \\frac{1}{f &#8216; (a ) } = \\frac{1}{f &#8216; (f^{-1}(b ))}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n<p>\uc774\ub7ec\ud55c \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \uc77c\ubc18\ud654\ub420 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem\">\n<p><span class=\"theorem\">\uc815\ub9ac 2. (\uc5ed\ud568\uc218 \ubbf8\ubd84\ubc95)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uc810 \\(a\\)\ub97c \uc6d0\uc18c\ub85c \uac16\ub294 \ud55c \uc5f4\ub9b0 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc77c\ub300\uc77c\uc774\uba70 \\(a\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f &#8216; (a) \\ne 0\\)\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uce58\uc5ed\uc744 \\(J\\)\ub77c\uace0 \ud558\uace0 \\(b = f(a)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f^{-1} : J \\to I\\)\ub294 \\(b\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0<br \/>\n\\[ (f^{-1})&#8217; (b) = \\frac{1}{f &#8216; (f^{-1} (b))}\\tag{10}\\]<br \/>\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \ub77c\uc774\ud504\ub2c8\uce20 \ud45c\uae30\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc774 \uc2dd\uc744 \ud45c\ud604\ud558\uba74<br \/>\n\\[\\left.\\frac{df^{-1}}{dx}\\right\\vert_{x=b} = \\frac{1}{\\left.\\frac{df}{dx}\\right\\vert_{x=f^{-1}(b)}}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85<\/p>\n<p><!-- p>\\(f\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uc77c\ub300\uc77c \ub300\uc751\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc21c\uc99d\uac00\ud558\uac70\ub098 \ub610\ub294 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc21c\uac10\uc18c\ud55c\ub2e4. \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc21c\uc99d\uac00\ud558\ub294 \ud568\uc218\ub77c\uace0 \ud558\uc790. \\(f ' (a) \\ne 0\\)\uc774\ubbc0\ub85c \\(f ' (a) > 0\\)\uc774\uac70\ub098 \\(f ' (a) < 0\\)\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(f\\)\uac00 \uc21c\uc99d\uac00\ud558\ubbc0\ub85c \\(f ' (a) > 0\\)\uc774\ub2e4.<\/p -->\n<p>\\(f^{-1}\\)\uac00 \\(J\\)\uc5d0\uc11c \uc77c\ub300\uc77c\uc774\ubbc0\ub85c \\(x \\ne b\\)\uc77c \ub54c \\(f^{-1}(x) \\ne f^{-1}(b)\\)\uc774\ub2e4. \ub354\uc6b1\uc774 \\(x \\ne b\\)\uc778 \uc784\uc758\uc758 \\(x\\in J\\)\uc5d0 \ub300\ud558\uc5ec \\(f(y)=x\\)\uc778 \\(y\\in I\\)\uac00 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud558\uba70, \\(f(y) \\ne f(a)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n(f^{-1})&#8217; (b)<br \/>\n&#038;= \\lim_{x\\to b}\\frac{f^{-1} (x) -f^{-1}(b)}{x-b} \\\\[6pt]<br \/>\n&#038;= \\lim_{y\\to a}\\frac{y-a}{f(y)-f(a)} \\\\[6pt]<br \/>\n&#038;= \\lim_{y\\to a}\\frac{1}{\\frac{f(y)-f(a)}{y-a}} \\\\[6pt]<br \/>\n&#038;= \\frac{1}{f &#8216; (a)} \\\\[6pt]<br \/>\n&#038;= \\frac{1}{f &#8216; ( f^{-1} (b))}. \\tag*{\\(\\blacksquare\\)}<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 5.<\/span><br \/>\n\uc9c0\uc218\ud568\uc218 \\(f(x)=e^x\\)\uc758 \ub3c4\ud568\uc218\uac00 \\(f &#8216; (x)=e^x\\)\uc774\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uc5ec \ub85c\uadf8\ud568\uc218 \\(f^{-1} (x) = \\ln x\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[<br \/>\n\\frac{d}{dx} \\ln x = (f^{-1}) &#8216; (x) = \\frac{1}{f &#8216; (f^{-1} (x))} = \\frac{1}{e^{\\ln x}} = \\frac{1}{x}.\\tag*{\\(\\square\\)}\\]\n<\/p>\n<\/div>\n<p>\ubcf4\uae30 5\uc758 \ub85c\uadf8\ud568\uc218\ub294 \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ubbf8\ubd84\ud560 \uc218\ub3c4 \uc788\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\ny &#038;= \\ln x, \\\\[6pt]<br \/>\ne^y &#038;= x, \\\\[6pt]<br \/>\n\\frac{d}{dx}(e^y) &#038;= \\frac{d}{dx}x,\\\\[6pt]<br \/>\ne^y \\frac{dy}{dx} &#038;= 1, \\\\[6pt]<br \/>\n\\frac{dy}{dx} &#038;= \\frac{1}{e^y} = \\frac{1}{x}.<br \/>\n\\end{align}\\]<br \/>\n\ud558\uc9c0\ub9cc \uc774 \ubc29\ubc95\uc5d0\ub294 \ud55c \uac00\uc9c0 \ubb38\uc81c\uc810\uc774 \uc788\ub2e4. \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc5ed\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uacbd\uc6b0 \uc5ed\ud568\uc218\uc758 \ub3c4\ud568\uc218\uc758 \uc2dd\uc744 \uad6c\ud560 \uc218 \uc788\uc744 \ubfd0 \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84 \uac00\ub2a5\uc131\uc740 \ubcf4\uc7a5\ud558\uc9c0 \uc54a\ub294\ub2e4\ub294 \uc810\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc5ed\ud568\uc218\ub97c \ubbf8\ubd84\ud560 \ub54c\uc5d0\ub294 \uc5ed\ud568\uc218\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c\uc9c0 \uc5ec\ubd80\ub3c4 \ub530\uc838\uc57c \ud55c\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.<\/span><br \/>\n\uc815\uc758\uc5ed\uc774 \\([-\\pi\/2 ,\\, \\pi\/2]\\)\uc77c \ub54c \\(f(x) = \\sin x\\)\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ub2e4. \\(f &#8216; (x) = \\cos x\\)\ub77c\ub294 \uc0ac\uc2e4\uc744 \uc774\uc6a9\ud558\uc5ec \\(f\\)\uc758 \uc5ed\ud568\uc218<br \/>\n\\[f^{-1} (x)= \\sin^{-1} x \\quad (-1\\le x \\le 1)\\]<br \/>\n\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \\(x=-\\pi\/2\\) \ub610\ub294 \\(x = \\pi\/2\\)\uc77c \ub54c \\(f &#8216; (x) = 0\\)\uc774\ubbc0\ub85c, \\(x=-1\\) \ub610\ub294 \\(x=1\\)\uc77c \ub54c \\(f^{-1}\\)\uc758 \ubbf8\ubd84\uacc4\uc218\ub294 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \\(-1 < x < 1\\)\uc77c \ub54c\uc5d0\ub294 \\(f^{-1}\\)\ub294 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70\n\\[\\begin{align}\n(f^{-1})'(x)\n&#038;= \\frac{1}{f ' (f^{-1} (x))} \\\\[6pt]\n&#038;= \\frac{1}{\\cos(\\sin^{-1} x)} \\\\[6pt]\n&#038;= \\frac{1}{\\sqrt{1-\\sin^2 (\\sin^{-1} x)}} \\\\[6pt]\n&#038;= \\frac{1}{\\sqrt{1-x^2}}\n\\end{align}\\]\n\uc989\n\\[\\frac{d}{dx}\\sin^{-1} x = \\frac{1}{\\sqrt{1-x^2}} \\quad (\\lvert x \\rvert < 1)\\]\n\uc774\ub2e4.\n<span class=\"qee\"><\/span><\/p>\n<\/div>\n<p><!--\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\uc608\uc81c n.<\/span>\n...\n<\/p>\n\n\n\n\n\n<p><span class=\"proof\">\ud480\uc774.<\/span>\n...\n<span class=\"qee\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"theorem\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac n.<\/span><\/p>\n\n\n\n\n<p>..<\/p>\n\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<span class=\"qed\"><\/span><\/p>\n\n\n<\/div>\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \ubbf8\ubd84 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uba74 \uc720\ub9ac\ud568\uc218\uc758 \ubbf8\ubd84\uc740 \ud560 \uc218 \uc788\uc9c0\ub9cc, \uadf8 \uc678\uc758 \ubcf5\uc7a1\ud55c \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uae30\uc5d0\ub294 \uc5b4\ub824\uc6c0\uc774 \uc788\ub2e4. \ud569\uc131\ud568\uc218, \uc74c\ud568\uc218, \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uba74 \ub354 \ub2e4\uc591\ud55c \uc885\ub958\uc758 \ud568\uc218\ub97c \ubbf8\ubd84\ud560 \uc218 \uc788\ub2e4. \ud569\uc131\ud568\uc218\uc758 \ubbf8\ubd84 \ub2e4\uc74c\uacfc \uac19\uc740 \ud568\uc218\ub97c \uc0dd\uac01\ud558\uc790. \\(h(x) = (2x+4)^3 \\) \uc774 \ud568\uc218\ub97c \ubbf8\ubd84\ud558\uace0 \ubbf8\ubd84\uacc4\uc218 \\(h &#8216; (1)\\)\uc744 \uad6c\ud574 \ubcf4\uc790. \uc6b0\ubcc0\uc744 \uc804\uac1c\ud558\uba74 \\(h(x) = 8x^3 + 48x^2 + 96x + 64 \\) \uc774\ubbc0\ub85c \\(h &#8216; (x)&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[210,199,215,212,216,162,217,198,218,213,219,209,214,211],"class_list":["post-1948","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-chain-rule","tag-differentiation-rule","tag-explicit-function","tag-implicit-differentiation","tag-implicit-function","tag-inverse-function-theorem","tag-217","tag-198","tag-218","tag-213","tag-219","tag-209","tag-214","tag-211"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1948","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"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=1948"}],"version-history":[{"count":55,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1948\/revisions"}],"predecessor-version":[{"id":2949,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1948\/revisions\/2949"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1948"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1948"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1948"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}