{"id":1952,"date":"2019-03-20T12:03:13","date_gmt":"2019-03-20T03:03:13","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1952"},"modified":"2021-12-14T19:25:29","modified_gmt":"2021-12-14T10:25:29","slug":"calculus-derivatives-of-exponential-and-logarithm-functions","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-derivatives-of-exponential-and-logarithm-functions\/","title":{"rendered":"\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84"},"content":{"rendered":"

\\(a\\)\uac00 \\(1\\)\uc774 \uc544\ub2cc \uc591\uc218\uc77c \ub54c
\n\\[ y = a^x \\,\\,\\,(x\\in\\mathbb{R})\\tag{1}\\]
\n\uaf34\ub85c \uc815\uc758\ub41c \ud568\uc218\ub97c \uc9c0\uc218\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uc9c0\uc218\ud568\uc218 (1)\uc758 \uc5ed\ud568\uc218\ub97c \ub85c\uadf8\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0
\n\\[ y = \\log _a x \\,\\,\\,(x > 0 )\\]
\n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uace0, \ub85c\uadf8 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc9c0\uc218\uac00 \uc2e4\uc218\uc778 \uac70\ub4ed\uc81c\uacf1 \ubbf8\ubd84\ubc95\uc744 \uc99d\uba85\ud55c\ub2e4.<\/p>\n

\uc9c0\uc218\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n

\uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc790\uc5f0\uc0c1\uc218<\/span>\ub77c\uace0 \ubd88\ub9ac\ub294 \uc0c1\uc218 \\(e\\)\ub97c \ub3c4\uc785\ud574\uc57c \ud55c\ub2e4. \\(e\\)\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec \uac00\uc9c0\uac00 \uc788\ub294\ub370, \uc5ec\uae30\uc11c\ub294 \ubbf8\ubd84\uacfc \uc801\ubd84\uc744 \ud558\uae30\uc5d0 \uac00\uc7a5 \uc720\uc6a9\ud55c \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud558\ub3c4\ub85d \ud558\uc790. \uc989 \\(a\\)\uac00 \uc591\uc218\uc77c \ub54c \uadf9\ud55c
\n\\[\\lim_{h\\to 0} \\frac{a^h -1}{h}\\tag{2}\\]
\n\uc740 \uc218\ub834\ud558\ub294\ub370, \uadf9\ud55c\uac12\uc740 \\(a\\)\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c4\ub2e4. \uc774\ub54c (2)\uc758 \uadf9\ud55c\uac12\uc774 \\(1\\)\uc774 \ub418\ub3c4\ub85d \ud558\ub294 \\(a\\)\uc758 \uac12\uc744 \\(e\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\n

\uc815\uc758 1. (\uc790\uc5f0\uc0c1\uc218)<\/span><\/p>\n

\ub4f1\uc2dd
\n\\[\\lim_{h\\to 0} \\frac{e^h -1}{h} = 1\\tag{3}\\]
\n\uc774 \uc131\ub9bd\ud558\ub3c4\ub85d \ud558\ub294 \uc2e4\uc218 \\(e\\)\ub294 \ub2e8 \ud558\ub098\uac00 \uc874\uc7ac\ud55c\ub2e4. \uadf8 \uac12\uc744 \uc790\uc5f0\uc0c1\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(e\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<\/div>\n

\uc790\uc5f0\uc0c1\uc218\ub294 \uadf8 \uac12\uc774
\n\\[e = 2.718281828459045 \\cdots\\]
\n\uc778 \ubb34\ub9ac\uc218\uc784\uc774 \uc54c\ub824\uc838 \uc788\ub2e4.<\/p>\n

\ud3b8\uc758\uc0c1 \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubc11\uc774 \uc790\uc5f0\uc0c1\uc218\uc778 \uc9c0\uc218\ud568\uc218\ub97c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub77c\uace0 \ubd80\ub974\uae30\ub85c \ud55c\ub2e4. \uc2dd (3)\uc744 \uc774\uc6a9\ud558\uba74 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc9c1\uc811 \uc5bb\uc744 \uc218 \uc788\ub2e4.
\n\\[\\begin{align}
\n\\frac{d}{dx}e^x
\n&= \\lim_{h\\to 0}\\frac{e^{x+h}-e^x}{h} \\\\[6pt]
\n&= \\lim_{h\\to 0}\\frac{e^x e^h – e^x}{h} \\\\[6pt]
\n&= \\lim_{h\\to 0}\\frac{e^x (e^h – 1)}{h} \\\\[6pt]
\n&= e^x \\cdot \\lim_{h\\to 0}\\frac{e^h – 1}{h} \\\\[8pt]
\n&= e^x \\cdot 1 = e^x .
\n\\end{align}\\]
\n\ubc11\uc774 \\(e\\)\uac00 \uc544\ub2cc \uc9c0\uc218\ud568\uc218\ub97c \ubbf8\ubd84\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc790\uc5f0\ub85c\uadf8\ud568\uc218\ub97c \ub3c4\uc785\ud574\uc57c \ud55c\ub2e4. \ubc11\uc774 \\(e\\)\uc778 \ub85c\uadf8\ud568\uc218\ub97c \uc790\uc5f0\ub85c\uadf8\ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub974\uace0 \\(\\ln\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[\\ln x = \\log_e x \\quad (x > 0)\\]
\n\uc774\uba70
\n\\[y = \\ln x \\quad \\Longleftrightarrow \\quad x=e^y \\quad ( x > 0 )\\]
\n\uc774\ub2e4. \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc640 \uc790\uc5f0\ub85c\uadf8\ud568\uc218\ub294 \uc11c\ub85c \uc5ed\ud568\uc218 \uad00\uacc4\uc774\ubbc0\ub85c \uc591\uc218 \\(t\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[e^{\\ln t} = t\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc5d0 \\(t = a^x,\\) \\(a > 0\\)\uc744 \ub300\uc785\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n\\[a^x = e^{\\ln a^x} = e^{x \\ln a}.\\tag{4}\\]\n<\/div>\n

\uc989 \ubaa8\ub4e0 \uc9c0\uc218\ud568\uc218\ub294 \ubc11\uc774 \\(e\\)\uc778 \uc9c0\uc218\ud568\uc218\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(a > 0\\)\uc77c \ub54c
\n\\[\\frac{d}{dx} a^x = \\frac{d}{dx} e^{x\\ln a} = e^{x\\ln a} \\cdot \\ln a = a^x \\ln a\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc774\ub85c\uc368 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\uc9c0\uc218\ud568\uc218\uc758 \ub3c4\ud568\uc218)<\/span><\/p>\n

\\[\\begin{align}
\n\\frac{d}{dx} e^x &= e^x , \\\\[6pt]
\n\\frac{d}{dx} a^x &= a^x \\ln a \\quad (a > 0).
\n\\end{align}\\]\n<\/p><\/div>\n

\ub85c\uadf8\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n

\ub85c\uadf8\ud568\uc218\ub294 \uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\uc774\ubbc0\ub85c \uc5ed\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \ub3c4\ud568\uc218\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uba3c\uc800 \\(f(y) =\\)\\(e^y\\)\ub77c\uace0 \ud558\uba74 \\(f^{-1} (x) =\\ln x\\)\uc774\ubbc0\ub85c
\n\\[\\begin{align}
\n\\frac{d}{dx} \\ln x
\n&= \\frac{d}{dx} f^{-1}(x) \\\\[6pt]
\n&= \\frac{1}{f ‘ (f^{-1}(x))}\\\\[6pt]
\n&= \\frac{1}{e^{f^{-1}(x)}}\\\\[6pt]
\n&= \\frac{1}{e^{\\ln x}} \\\\[6pt]
\n&= \\frac{1}{x}
\n\\end{align}\\]
\n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(g(y) = a^y ,\\) \\(a > 0,\\) \\(a\\ne 1\\)\uc774\ub77c\uace0 \ud558\uba74 \\(g^{-1} (x) = \\log_a x\\)\uc774\ubbc0\ub85c
\n\\[\\begin{align}
\n\\frac{d}{dx} \\log_a x
\n&= \\frac{d}{dx} g^{-1}(x) \\\\[6pt]
\n&= \\frac{1}{g ‘ (g^{-1}(x))}\\\\[6pt]
\n&= \\frac{1}{a^{f^{-1}(x)}\\cdot \\ln a}\\\\[6pt]
\n&= \\frac{1}{a^{\\log_a x}\\cdot \\ln a} \\\\[6pt]
\n&= \\frac{1}{x \\ln a}
\n\\end{align}\\]
\n\uc774\ub2e4. \uc774\ub85c\uc368 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2. (\ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218)<\/span><\/p>\n

\\[\\begin{align}
\n\\frac{d}{dx} \\ln x &= \\frac{1}{x}, \\\\[6pt]
\n\\frac{d}{dx} \\log_a x &= \\frac{1}{x \\ln a} \\quad ( a > 0,\\, a\\ne 1).
\n\\end{align}\\]\n<\/p><\/div>\n

\uc790\uc5f0\uc0c1\uc218\ub294 \ub2e4\ub978 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc815\ub9ac 3. (\uc790\uc5f0\uc0c1\uc218\uc758 \ub2e4\ub978 \uc815\uc758)<\/span><\/p>\n

\\[e = \\lim_{x\\to 0} (1+x)^{1\/x}.\\tag{5}\\]\n<\/p><\/div>\n

\n

\uc99d\uba85<\/p>\n

\\(f(x) = \\ln x\\)\ub77c\uace0 \ud558\uc790. \\(f ‘ (x) = 1\/x\\)\uc774\ubbc0\ub85c \\(f ‘ (1)=1\\)\uc774\ub2e4. \ud55c\ud3b8 \ubbf8\ubd84\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec
\n\\[\\begin{align}
\nf ‘ (1)
\n&= \\lim_{h\\to 0}\\frac{f(1+h)-f(1)}{h}\\\\[6pt]
\n&= \\lim_{x\\to 0}\\frac{f(1+x)-f(1)}{x}\\\\[6pt]
\n&= \\lim_{x\\to 0}\\frac{\\ln (1+x) – \\ln 1}{x} \\\\[6pt]
\n&= \\lim_{x\\to 0}\\frac{\\ln (1+x)}{x}\\\\[6pt]
\n&= \\lim_{x\\to 0}\\ln (1+x)^{1\/x}\\\\[6pt]
\n&= \\ln\\left[ \\lim_{x\\to 0} (1+x)^{1\/x} \\right].
\n\\end{align}\\]
\n\uadf8\ub7f0\ub370 \\(f ‘ (1)=1\\)\uc774\ubbc0\ub85c \uc704 \uc2dd\uc758 \uac12\uc740 \\(1\\)\uc774 \ub418\uc5b4\uc57c \ud55c\ub2e4. \uc989
\n\\[\\ln\\left[ \\lim_{x\\to 0} (1+x)^{1\/x} \\right] = 1.\\]
\n\uc704 \ub4f1\uc2dd\uc758 \uc88c\ubcc0\uc5d0\uc11c \uc790\uc5f0\ub85c\uadf8\ud568\uc218\ub294 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\uc774\ubbc0\ub85c
\n\\[\\lim_{x\\to 0}(1+x)^{1\/x} = e\\]
\n\ub97c \uc5bb\ub294\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\ub530\ub984\uc815\ub9ac 1. (\uc790\uc5f0\uc0c1\uc218\uc758 \ub2e4\ub978 \uc815\uc758)<\/span><\/p>\n

\\[e = \\lim_{x\\to \\infty} \\left( 1+ \\frac{1}{x} \\right) ^x .\\tag{6}\\]\n<\/p><\/div>\n

\n

\uc99d\uba85<\/p>\n

\uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec
\n\\[\\lim_{t\\to 0^+} (1+t)^{1\/t} = e\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(x = 1\/t\\)\uc774\ub77c\uace0 \ud558\uba74 \\(t\\to 0^+\\)\uc77c \ub54c \\(x\\to \\infty\\)\uc774\ubbc0\ub85c \uc815\ub9ac 3\uc5d0 \uc758\ud558\uc5ec
\n\\[\\lim_{x\\to \\infty} \\left( 1+ \\frac{1}{x} \\right) ^x = e\\]
\n\uac00 \uc131\ub9bd\ud55c\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\uc0ac\uc2e4 \uc790\uc5f0\uc0c1\uc218\ub97c \uc815\ub9ac 3\uc774\ub098 \ub530\ub984\uc815\ub9ac 1\uc5d0\uc11c\uc640 \uac19\uc774 \u2018\uc815\uc758\u2019\ud574\ub3c4 \ub41c\ub2e4. \uc774\uc640 \uac19\uc774 \uc815\uc758\ud574\ub3c4 \ub4f1\uc2dd (4), \uc815\ub9ac 1, \uc815\ub9ac 2\ub97c \ubaa8\ub450 \uc720\ub3c4\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\ud55c\ud3b8 \uc790\uc5f0\ub85c\uadf8\ud568\uc218\ub97c \uba3c\uc800 \uc815\uc758\ud55c \ub4a4 \ub2e4\ub978 \ubaa8\ub4e0 \uac83\uc744 \ub04c\uc5b4\ub0b4\ub294 \ubc29\ubc95\ub3c4 \uc788\ub2e4. \uc989 \uc790\uc5f0\ub85c\uadf8\ud568\uc218\ub97c
\n\\[\\ln x := \\int_1^x \\frac{1}{t} dt \\quad (x > 0)\\]
\n\uc774\ub77c\uace0 \uc815\uc758\ud558\uace0, \uc790\uc5f0\uc9c0\uc218\ud568\uc218 \\(\\exp (x)\\)\ub97c \uc790\uc5f0\ub85c\uadf8\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub85c \uc815\uc758\ud55c \ub4a4 \\(e = \\exp(1)\\)\ub85c \uc815\uc758\ud558\ub294 \ubc29\ubc95\ub3c4 \uc788\ub2e4.<\/p>\n

\uac70\ub4ed\uc81c\uacf1\uae09\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub97c \uba3c\uc800 \uc815\uc758\ud558\ub294 \ubc29\ubc95\ub3c4 \uc788\ub2e4. \uc989 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub97c
\n\\[\\exp (x) := \\sum_{n=0}^{\\infty} \\frac{x^n}{n!} \\quad (x\\in\\mathbb{R})\\]
\n\ub77c\uace0 \uc815\uc758\ud558\uace0, \uc790\uc5f0\ub85c\uadf8\ud568\uc218 \\(\\ln x\\)\ub97c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub85c \uc815\uc758\ud558\ub294 \ubc29\ubc95\ub3c4 \uc788\ub2e4.<\/p>\n

\uc790\uc5f0\uc0c1\uc218\uc640 \uc790\uc5f0\ub85c\uadf8\uc758 \uc815\uc758, \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc720\ub3c4\ud558\ub294 \uacfc\uc815\uc740 \ucc45\ub9c8\ub2e4 \ub2e4\ub974\uc9c0\ub9cc \uadf8 \uacb0\uacfc\ub294 \ubaa8\ub450 \ub3d9\uce58\uc774\ub2e4.<\/p>\n

\ub85c\uadf8 \ubbf8\ubd84\ubc95<\/h3>\n

\ud568\uc218 \\(\\ln \\lvert x \\rvert\\)\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \\(x > 0\\)\uc77c \ub54c\uc5d0\ub294
\n\\[\\frac{d}{dx} \\ln \\lvert x \\rvert = \\frac{d}{dx} \\ln x = \\frac{1}{x}\\]
\n\uc774\uba70, \\(x < 0\\)\uc77c \ub54c\uc5d0\ub294\n\\[\\frac{d}{dx} \\ln \\lvert x \\rvert = \\frac{d}{dx} \\ln (-x) = \\frac{1}{-x} \\cdot (-1) = \\frac{1}{x} \\]\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n\\[\\frac{d}{dx} \\ln \\lvert x \\rvert = \\frac{1}{x} \\quad (x\\ne 0)\\tag{7}\\]\n<\/div>\n

\ub85c\uadf8\ud568\uc218\ub97c \uc774\uc6a9\ud558\uba74 \ubcf5\uc7a1\ud55c \ubd84\uc218\uc2dd\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uc27d\uac8c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\ubcf4\uae30 1.<\/span>
\n\ub2e4\uc74c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.
\n\\[f(x) = \\frac{(x^2 +1)\\sqrt{x+2}}{x+3} \\quad (x \\ge -2).\\]
\n\uc774 \ud568\uc218\ub294 \uc815\uc758\uc5ed\uc758 \uc810 \uc911\uc5d0\uc11c \\(-2\\)\uac00 \uc544\ub2cc \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \\(f(x)=y\\)\ub77c\uace0 \ud558\uace0 \uc704 \uc2dd\uc758 \uc591\ubcc0\uc5d0 \uc790\uc5f0\ub85c\uadf8\ub97c \ucde8\ud558\uba74
\n\\[\\ln \\lvert y \\rvert = \\ln \\left\\lvert \\frac{(x^2 +1)(x+2)^{1\/2}}{x+3} \\right\\rvert \\quad (x > -2)\\]
\n\uc774\ub2e4. \ub85c\uadf8\uc758 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\uc5ec \uc6b0\ubcc0\uc744 \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c \uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\ln\\lvert y \\rvert = \\ln \\lvert x^2 +1 \\rvert + \\frac{1}{2} \\ln \\lvert x+2 \\rvert – \\ln \\lvert x+3 \\rvert \\quad (x > -2).\\]
\n\uc591\ubcc0\uc744 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uba74
\n\\[\\frac{y ‘}{y} = \\frac{2x}{x^2 +1} + \\frac{1}{2(x+2)} – \\frac{1}{x+3} \\quad (x > -2)\\]
\n\uc774\uace0, \uc774 \uc2dd\uc744 \\(y ‘ \\)\uc5d0 \ub300\ud558\uc5ec \ud480\uba74
\n\\[\\begin{align}
\ny ‘
\n&= y \\left( \\frac{2x}{x^2 +1} + \\frac{1}{2(x+2)} – \\frac{1}{x+3} \\right) \\\\[6pt]
\n&= \\frac{(x^2 +1)\\sqrt{x+2}}{x+3} \\left( \\frac{2x}{x^2 +1} + \\frac{1}{2(x+2)} – \\frac{1}{x+3} \\right) \\quad (x > -2)
\n\\end{align}\\]
\n\uc744 \uc5bb\ub294\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\uc704 \ubcf4\uae30\uc5d0\uc11c\uc640 \uac19\uc774 \uc591\ubcc0\uc5d0 \uc790\uc5f0\ub85c\uadf8\ub97c \ucde8\ud55c \ub4a4 \ubbf8\ubd84\ud558\ub294 \ubc29\ubc95\uc744 \ub85c\uadf8 \ubbf8\ubd84\ubc95<\/span>(logarithmic differentiation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\n

\ubcf4\uae30 2.<\/span>
\n\ub2e4\uc74c \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.
\n\\[f(x) = x^x \\quad (x > 0).\\]
\n\uc591\ubcc0\uc5d0 \uc790\uc5f0\ub85c\uadf8\ub97c \ucde8\ud558\uba74
\n\\[\\ln f(x) = x\\ln x\\]
\n\uc774\uba70, \uc591\ubcc0\uc744 \\(x\\)\uc5d0 \uad00\ud558\uc5ec \ubbf8\ubd84\ud558\uba74
\n\\[\\frac{f ‘ (x)}{f(x)} = \\ln x + \\frac{x}{x}\\]
\n\uc989
\n\\[\\frac{f ‘ (x)}{x^x} = \\ln x + 1\\]
\n\uc774\ub2e4. \uc774 \uc2dd\uc744 \\(f ‘ (x)\\)\uc5d0 \ub300\ud558\uc5ec \ud480\uba74
\n\\[f ‘ (x) = x^x(\\ln x + 1)\\]
\n\uc774\ub2e4.<\/p>\n

\ud568\uc218 \\(f(x)=x^x\\)\uc740 \ub2e4\ubcc0\uc218\ud568\uc218\uc758 \uc5f0\uc1c4 \ubc95\uce59\uc744 \uc774\uc6a9\ud558\uc5ec \ubbf8\ubd84\ud560 \uc218\ub3c4 \uc788\ub2e4.
\n\\[\\phi(s,\\,t) = s^t ,\\,\\,\\,s=x,\\,\\,\\,t=x\\]
\n\ub77c\uace0 \ud558\uba74 \\(f(x) = \\phi(s,\\,t)\\)\uc774\ubbc0\ub85c
\n\\[\\begin{align}
\n\\frac{df}{dx}
\n&= \\frac{d\\phi}{dx} = \\frac{\\partial \\phi}{\\partial s} \\frac{\\partial s}{\\partial x} + \\frac{\\partial \\phi}{\\partial t} \\frac{\\partial t}{\\partial x}\\\\[6pt]
\n&= ts^{t-1} \\cdot 1 + s^t \\ln s \\cdot 1\\\\[8pt]
\n&= x^x + x^x \\ln x \\\\[8pt]
\n&= x^x (\\ln x + 1)
\n\\end{align}\\]
\n\uc744 \uc5bb\ub294\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\ub85c\uadf8 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc2e4\uc218 \uc9c0\uc218\uc5d0 \ub300\ud55c \uac70\ub4ed\uc81c\uacf1 \ubbf8\ubd84\ubc95\uce59\uc744 \uc99d\uba85\ud558\uc790.<\/p>\n

\n

\uc815\ub9ac 4. (\uc2e4\uc218 \uc9c0\uc218\uc5d0 \ub300\ud55c \uac70\ub4ed\uc81c\uacf1 \ubbf8\ubd84 \ubc95\uce59)<\/span><\/p>\n

\\(x > 0\\)\uc774\uace0 \\(\\alpha\\)\uac00 \uc2e4\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
\n\\[\\frac{d}{dx} x^{\\alpha} = \\alpha x^{\\alpha -1}.\\tag{8}\\]
\n\\(x \\le 0\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(x^{\\alpha}\\)\uacfc \\(x^{\\alpha -1}\\)\uc774 \ubaa8\ub450 \uc815\uc758\ub418\uace0 (8)\uc758 \uc88c\ubcc0\uc758 \ubbf8\ubd84\uc774 \uc874\uc7ac\ud558\ub294 \ubaa8\ub4e0 \uacbd\uc6b0\uc5d0 \ub300\ud558\uc5ec \ub4f1\uc2dd (8)\uc774 \uc131\ub9bd\ud55c\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\uba3c\uc800 \\(x > 0\\)\uc77c \ub54c\uc5d0\ub294 \\(x^{\\alpha}\\)\ub97c \\(x\\)\uc5d0 \ub300\ud558\uc5ec \ubbf8\ubd84\ud558\uba74 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\frac{d}{dx} x^{\\alpha}
\n&= \\frac{d}{dx} e^{\\alpha \\ln x} \\\\[6pt]
\n&= e^{\\alpha \\ln x} \\cdot \\frac{d}{dx} (\\alpha \\ln x )\\\\[6pt]
\n&= x^{\\alpha} \\cdot \\frac{\\alpha}{x} \\\\[6pt]
\n&= \\alpha x^{\\alpha -1}.
\n\\end{align}\\]
\n\ub2e4\uc74c\uc73c\ub85c \\(x < 0\\)\uc77c \ub54c\ub97c \uc99d\uba85\ud558\uc790. \\(y=x^{\\alpha},\\) \\(y ' ,\\) \\(x^{\\alpha -1}\\)\uc774 \uc815\uc758\ub418\ub294 \\(x,\\) \\(\\alpha\\)\uc5d0 \ub300\ud558\uc5ec\n\\[\\ln \\lvert y \\rvert = \\ln \\lvert x \\rvert^{\\alpha} = \\alpha \\ln \\lvert x \\rvert\\]\n\uc774\ubbc0\ub85c \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.\n\\[\\frac{y ' }{y} = \\frac{\\alpha}{x}.\\]\n\uc774 \uc2dd\uc744 \\(y ' \\)\uc5d0 \ub300\ud558\uc5ec \ud480\uba74\n\\[y ' = \\alpha \\frac{y}{x} = \\alpha \\frac{x^{\\alpha}}{x} = \\alpha x^{\\alpha -1}\\]\n\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\ub05d\uc73c\ub85c \\(x=0,\\) \\(\\alpha \\ge 1\\)\uc77c \ub54c\uc5d0\ub294 \ubbf8\ubd84\uacc4\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(y ‘ =0\\)\uc774\ub2e4.<\/p>\n

\uc774\ub85c\uc368 \ubaa8\ub4e0 \uacbd\uc6b0\uc5d0 \ub300\ud558\uc5ec (8)\uc774 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc600\ub2e4.<\/span><\/p>\n<\/div>\n

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

\\(a\\)\uac00 \\(1\\)\uc774 \uc544\ub2cc \uc591\uc218\uc77c \ub54c \\( y = a^x \\,\\,\\,(x\\in\\mathbb{R})\\) \uaf34\ub85c \uc815\uc758\ub41c \ud568\uc218\ub97c \uc9c0\uc218\ud568\uc218\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uc9c0\uc218\ud568\uc218 (1)\uc758 \uc5ed\ud568\uc218\ub97c \ub85c\uadf8\ud568\uc218\ub77c\uace0 \ubd80\ub974\uace0 \\( y = \\log _a x \\,\\,\\,(x > 0 )\\) \ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uace0, \ub85c\uadf8 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc9c0\uc218\uac00 \uc2e4\uc218\uc778 \uac70\ub4ed\uc81c\uacf1 \ubbf8\ubd84\ubc95\uc744 \uc99d\uba85\ud55c\ub2e4. \uc9c0\uc218\ud568\uc218\uc758 \ubbf8\ubd84 \uc9c0\uc218\ud568\uc218\uc640 \ub85c\uadf8\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uae30 \uc704\ud574\uc11c\ub294 \uc790\uc5f0\uc0c1\uc218\ub77c\uace0 \ubd88\ub9ac\ub294 \uc0c1\uc218 \\(e\\)\ub97c \ub3c4\uc785\ud574\uc57c \ud55c\ub2e4. \\(e\\)\ub97c \uc815\uc758\ud558\ub294 \ubc29\ubc95\uc740 \uc5ec\ub7ec…<\/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":[84,189,225,226,227,224,228,230,229,223],"class_list":["post-1952","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-derivative","tag-differentiation","tag-exponential-function","tag-logarithmic-function","tag-natural-logarithm","tag-224","tag-228","tag-230","tag-229","tag-223"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1952","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=1952"}],"version-history":[{"count":43,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1952\/revisions"}],"predecessor-version":[{"id":8333,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1952\/revisions\/8333"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1952"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1952"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1952"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}