{"id":1954,"date":"2019-03-21T12:04:21","date_gmt":"2019-03-21T03:04:21","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1954"},"modified":"2024-09-02T10:17:11","modified_gmt":"2024-09-02T01:17:11","slug":"calculus-derivatives-of-hyperbolic-functions","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-derivatives-of-hyperbolic-functions\/","title":{"rendered":"\uc30d\uace1\uc120\ud568\uc218\uc640 \uc5ed\uc30d\uace1\uc120\ud568\uc218\uc758 \ubbf8\ubd84"},"content":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc30d\uace1\uc120\ud568\uc218\uc640 \uc5ed\uc30d\uace1\uc120\ud568\uc218\ub97c \uc815\uc758\ud558\uace0 \uc774 \ud568\uc218\ub4e4\uc758 \ub3c4\ud568\uc218\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\uc30d\uace1\uc120\ud568\uc218\uc758 \uc815\uc758<\/h3>\n

\\(\\mathbb{R}\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub294 \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ud569\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc989 \ud568\uc218 \\(f\\)\ub294
\n\\[f(x) = \\frac{f(x)-f(-x)}{2} + \\frac{f(x)+f(-x)}{2}\\]
\n\ub85c\uc11c \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub97c \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ud558\uba74
\n\\[e^x = \\frac{e^x – e^{-x}}{2} + \\frac{e^x + e^{-x}}{2}\\]
\n\uc774\ub2e4. \uc774\ub54c \\(e^x\\)\uc758 \uae30\ud568\uc218 \ubd80\ubd84\uc744 \uc30d\uace1\uc120\uc0ac\uc778, \uc6b0\ud568\uc218 \ubd80\ubd84\uc744 \uc30d\uace1\uc120\ucf54\uc0ac\uc778\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \uc30d\uace1\uc120\uc0ac\uc778<\/span>(hyperbolic sine)\uc774\ub780
\n\\[\\sinh x = \\frac{e^x – e^{-x}}{2}\\]
\n\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(\\sinh\\)\ub97c \uc774\ub974\uba70, \uc30d\uace1\uc120\ucf54\uc0ac\uc778<\/span>(hyperbolic cosine)\uc774\ub780
\n\\[\\cosh x = \\frac{e^x + e^{-x}}{2}\\]
\n\uc73c\ub85c \uc815\uc758\ub41c \ud568\uc218 \\(\\cosh\\)\ub97c \uc774\ub978\ub2e4. \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\ub978 \ub124 \uac1c\uc758 \uc0bc\uac01\ud568\uc218\ub97c \ubaa8\ub450 \ud45c\ud604\ud560 \uc218 \uc788\ub294 \uac83\ucc98\ub7fc \uc30d\uace1\uc120\uc0ac\uc778\uacfc \uc30d\uace1\uc120\ucf54\uc0ac\uc778\uc744 \uc774\uc6a9\ud558\uc5ec \ub2e4\ub978 \ub124 \uac1c\uc758 \uc30d\uace1\uc120\ud568\uc218\ub97c \ubaa8\ub450 \ud45c\ud604\ud560 \uc218 \uc788\ub2e4. \uc989 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n

\\[
\n\\newcommand{sech}[]{\\operatorname{sech}}
\n\\newcommand{csch}[]{\\operatorname{csch}}
\n\\begin{align}
\n\\tanh x &= \\frac{\\sinh x}{\\cosh x} = \\frac{e^x – e^{-x}}{e^x + e^{-x}} ,\\\\[6pt]
\n\\coth x &= \\frac{\\cosh x}{\\sinh x} = \\frac{e^x + e^{-x}}{e^x – e^{-x}} ,\\\\[6pt]
\n\\sech x &= \\frac{1}{\\cosh x} = \\frac{2}{e^x + e^{-x}} ,\\\\[6pt]
\n\\csch x &= \\frac{1}{\\sinh x} = \\frac{2}{e^x – e^{-x}} .
\n\\end{align}
\n\\]<\/p>\n

\uc704 \ub124 \uac1c\uc758 \ud568\uc218\ub97c \uc21c\uc11c\ub300\ub85c \uc30d\uace1\uc120\ud0c4\uc820\ud2b8<\/span>, \uc30d\uace1\uc120\ucf54\ud0c4\uc820\ud2b8<\/span>, \uc30d\uace1\uc120\uc2dc\ucee8\ud2b8<\/span>, \uc30d\uace1\uc120\ucf54\uc2dc\ucee8\ud2b8<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \uadf8\ub9ac\uace0 \uc9c0\uae08\uae4c\uc9c0 \uc18c\uac1c\ud55c \uc5ec\uc12f \uac1c\uc758 \ud568\uc218\ub97c \ud1b5\ud2c0\uc5b4 \uc30d\uace1\uc120\ud568\uc218<\/span>(hyperbolic function)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uc30d\uace1\uc120\ud568\uc218\uc5d0 \uc0bc\uac01\ud568\uc218\uc758 \uc774\ub984\uc774 \ubd99\uc740 \uc774\uc720\ub294 \uc0ac\uc778\uacfc \ucf54\uc0ac\uc778\uc774 \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ub300\ud45c\uc801\uc778 \ud568\uc218\uc774\uae30\ub3c4 \ud558\uac70\ub2c8\uc640, \uc30d\uace1\uc120\ud568\uc218\uac00 \uc0bc\uac01\ud568\uc218\uc640 \ube44\uc2b7\ud55c \uc131\uc9c8\uc744 \uac00\uc9c0\uace0 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

\n

\uc30d\uace1\uc120\ud568\uc218\uc640 \uad00\ub828\ub41c \ud56d\ub4f1\uc2dd<\/span><\/p>\n

\\[\\begin{align}
\n\\cosh^2 x &- \\sinh^2 x = 1 ,\\\\[8pt]
\n\\sinh 2x &= 2 \\sinh x \\cosh x ,\\\\[8pt]
\n\\cosh 2x &= \\cosh^2 x + \\sinh^2 x ,\\\\[6pt]
\n\\cosh^2 x &= \\frac{\\cosh 2x +1}{2} ,\\\\[4pt]
\n\\sinh^2 x &= \\frac{\\cosh 2x -1}{2} ,\\\\[6pt]
\n\\tanh^2 x &= 1- \\sech^2 x ,\\\\[8pt]
\n\\coth^2 x &= 1+ \\csch^2 x .
\n\\end{align}\\]\n<\/p><\/div>\n

\uc5ec\uae30\uc11c \uccab \ubc88\uc9f8 \ub4f1\uc2dd\uc744 \uc720\uc2ec\ud788 \uc0b4\ud3b4\ubcfc \ud544\uc694\uac00 \uc788\ub2e4. \uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \\((\\cosh t ,\\, \\sinh t )\\)\ub85c \ud45c\ud604\ub418\ub294 \uc810\uc758 \uc88c\ud45c\ub294 \\(x^2 – y^2 = 1\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ubbc0\ub85c, \uc774 \uc810\uc740 \uc30d\uace1\uc120 \uc704\uc5d0 \ub193\uc774\uac8c \ub41c\ub2e4. \uc30d\uace1\uc120\ud568\uc218\uc758 \uc774\ub984\uc5d0 \u2018\uc30d\uace1\uc120\u2019\uc774\ub77c\ub294 \uc811\ub450\uc0ac\uac00 \ubd99\ub294 \uac83\uc740 \ubc14\ub85c \uc774 \ub54c\ubb38\uc774\ub2e4.<\/p>\n

\uc30d\uace1\uc120\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n

\uc30d\uace1\uc120\ud568\uc218\ub294 \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub97c \uc774\uc6a9\ud558\uc5ec \uc815\uc758\ub418\uae30 \ub54c\ubb38\uc5d0, \uc30d\uace1\uc120\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uac83\uc740 \uc27d\ub2e4.
\n\\[\\begin{align}
\n\\frac{d}{dx} \\sinh x &= \\frac{d}{dx} \\frac{e^x – e^{-x}}{2} \\\\[4pt]&= \\frac{e^x + e^{-x}}{2} = \\cosh x,\\\\[10pt]
\n\\frac{d}{dx} \\cosh x &= \\frac{d}{dx} \\frac{e^x + e^{-x}}{2} \\\\[4pt]&= \\frac{e^x – e^{-x}}{2} = \\sinh x,\\\\[10pt]
\n\\frac{d}{dx} \\tanh x &= \\frac{d}{dx} \\frac{\\sinh x}{\\cosh x} = \\frac{\\cosh^2 x – \\sinh^2}{(\\cosh x)^2} \\\\[4pt]&= \\frac{1}{(\\cosh x)^2} = \\sech^2 x ,\\\\[10pt]
\n\\frac{d}{dx} \\coth x &= \\frac{d}{dx} \\frac{\\cosh x}{\\sinh x} = \\frac{\\sinh^2 x – \\cosh^2}{(\\sinh x)^2} \\\\[4pt]&= \\frac{-1}{(\\sinh x)^2} = -\\csch^2 x \\,\\,\\, (x \\ne 0),\\\\[10pt]
\n\\frac{d}{dx} \\sech x &= \\frac{d}{dx} \\frac{1}{\\cosh x} = \\frac{-\\sinh x}{(\\cosh x)^2} \\\\[4pt]&= – \\sech x \\tanh x ,\\\\[10pt]
\n\\frac{d}{dx} \\csch x &= \\frac{d}{dx} \\frac{1}{\\sinh x} = \\frac{-\\cosh x}{(\\sinh x)^2} \\\\[4pt]&= – \\csch x \\coth x \\,\\,\\, (x \\ne 0).
\n\\end{align}\\]
\n\uc774\uac83\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\uc30d\uace1\uc120\ud568\uc218\uc758 \ub3c4\ud568\uc218)<\/span><\/p>\n

\\[\\begin{align}
\n\\frac{d}{dx} \\sinh x &= \\cosh x,\\\\[6pt]
\n\\frac{d}{dx} \\cosh x &= \\sinh x,\\\\[6pt]
\n\\frac{d}{dx} \\tanh x &= \\sech^2 x ,\\\\[6pt]
\n\\frac{d}{dx} \\coth x &= -\\csch^2 x &(&x \\ne 0),\\\\[6pt]
\n\\frac{d}{dx} \\sech x &= – \\sech x \\tanh x ,\\\\[6pt]
\n\\frac{d}{dx} \\csch x &= – \\csch x \\coth x &(&x \\ne 0).
\n\\end{align}\\]\n<\/p><\/div>\n

\uc5ed\uc30d\uace1\uc120\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n

\uc30d\uace1\uc120\uc0ac\uc778, \uc30d\uace1\uc120\ud0c4\uc820\ud2b8, \uc30d\uace1\uc120\ucf54\ud0c4\uc820\ud2b8, \uc30d\uace1\uc120\ucf54\uc2dc\ucee8\ud2b8\ub294 \uc77c\ub300\uc77c \ud568\uc218\uc774\ubbc0\ub85c \uadf8 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uadf8\ub7ec\ub098 \uc30d\uace1\uc120\ucf54\uc0ac\uc778\uacfc \uc30d\uace1\uc120\uc2dc\ucee8\ud2b8\ub294 \uc77c\ub300\uc77c \ud568\uc218\uac00 \uc544\ub2c8\ubbc0\ub85c \uadf8 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc5c6\ub2e4.<\/p>\n

\ud558\uc9c0\ub9cc \uc0bc\uac01\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc744 \ucd95\uc18c\ud558\uc5ec \uc77c\ub300\uc77c \ud568\uc218\uac00 \ub418\ub3c4\ub85d \ud558\uace0 \uadf8 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud55c \uac83\ucc98\ub7fc \uc30d\uace1\uc120\ucf54\uc0ac\uc778\uacfc \uc30d\uace1\uc120\uc2dc\ucee8\ud2b8\uc758 \uc815\uc758\uc5ed\uc744 \ucd95\uc18c\ud558\uace0 \uadf8 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\uc774\ub7ec\ud55c \uad00\uc810\uc5d0\uc11c \uc5ed\uc30d\uace1\uc120\ud568\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.<\/p>\n

\\[\\begin{align}
\ny = \\sinh^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x = \\sinh y \\,\\,\\,\\text{for}\\,\\,\\,y\\in\\mathbb{R} ,\\\\[8pt]
\ny = \\cosh^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x = \\cosh y \\,\\,\\,\\text{for}\\,\\,\\,y\\in [0,\\,\\infty ) ,\\\\[8pt]
\ny = \\tanh^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x = \\tanh y \\,\\,\\,\\text{for}\\,\\,\\,y\\in\\mathbb{R} ,\\\\[8pt]
\ny = \\coth^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x = \\coth y \\,\\,\\,\\text{for}\\,\\,\\,y\\in\\mathbb{R} \\setminus \\left\\{ 0 \\right\\},\\\\[8pt]
\ny = \\sech^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x = \\sech y \\,\\,\\,\\text{for}\\,\\,\\,y\\in [0,\\,\\infty ) ,\\\\[8pt]
\ny = \\csch^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x = \\csch y \\,\\,\\,\\text{for}\\,\\,\\,y\\in\\mathbb{R} \\setminus \\left\\{ 0 \\right\\}.
\n\\end{align}\\]<\/p>\n

\uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc5ed\uc30d\uace1\uc120\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc815\uc758\uc5ed\uc744 \uac00\uc9c4\ub2e4.<\/p>\n

\\[\\begin{align}
\ny = \\sinh^{-1} x \\,\\,\\,&\\text{for}\\,\\,\\,x\\in\\mathbb{R} ,\\\\[8pt]
\ny = \\cosh^{-1} x \\,\\,\\,&\\text{for}\\,\\,\\,x\\in [1,\\,\\infty ) ,\\\\[8pt]
\ny = \\tanh^{-1} x \\,\\,\\,&\\text{for}\\,\\,\\,x\\in (-1,\\,1) ,\\\\[8pt]
\ny = \\coth^{-1} x \\,\\,\\,&\\text{for}\\,\\,\\,x\\in\\mathbb{R} \\setminus [-1,\\,1],\\\\[8pt]
\ny = \\sech^{-1} x \\,\\,\\,&\\text{for}\\,\\,\\,x\\in (0,\\,1] ,\\\\[8pt]
\ny = \\csch^{-1} x \\,\\,\\,&\\text{for}\\,\\,\\,x\\in\\mathbb{R} \\setminus \\left\\{ 0 \\right\\}.
\n\\end{align}\\]<\/p>\n

\uc30d\uace1\uc120\ud568\uc218\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \\(0 < x \\le 1\\)\uc77c \ub54c\n\\[\\sech \\left( \\cosh^{-1} \\left( \\frac{1}{x} \\right) \\right)\n= \\frac{1}{\\cosh \\left( \\cosh^{-1} \\left( \\frac{1}{x} \\right) \\right)} = \\frac{1}{\\left(\\frac{1}{x}\\right)} = x\\]\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(\\sech (\\sech^{-1} x ) =x\\)\uc774\ubbc0\ub85c\n\\[\\cosh^{-1} \\left( \\frac{1}{x} \\right) = \\sech^{-1} x\\]\n\ub97c \uc5bb\ub294\ub2e4. \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc5ed\uc30d\uace1\uc120\ud568\uc218\uc640 \uad00\ub828\ub41c \ud56d\ub4f1\uc2dd<\/span><\/p>\n

\\[\\begin{align}
\n\\sech^{-1} x &= \\cosh^{-1} \\frac{1}{x}, \\\\[6pt]
\n\\csch^{-1} x &= \\sinh^{-1} \\frac{1}{x}, \\\\[6pt]
\n\\coth^{-1} x &= \\tanh^{-1} \\frac{1}{x} .
\n\\end{align}\\]\n<\/p><\/div>\n

\uc774\uc81c \uc5ed\uc30d\uace1\uc120\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n

\\[\\begin{align}<\/p>\n

\\frac{d}{dx} (\\sinh^{-1} x )
\n&=\\frac{1}{\\cosh (\\sinh^{-1} x)} \\\\[4pt]
\n&=\\frac{1}{\\sqrt{1+\\sinh^2 (\\sinh^{-1} x)}} \\\\[4pt]
\n&=\\frac{1}{\\sqrt{1+x^2}},\\\\[8pt]<\/p>\n

\\frac{d}{dx} (\\cosh^{-1} x )
\n&=\\frac{1}{\\sinh (\\cosh^{-1} x)} \\\\[4pt]
\n&=\\frac{1}{\\sqrt{\\cosh^2 (\\cosh^{-1} x)}-1} \\\\[4pt]
\n&=\\frac{1}{\\sqrt{x^2 -1}} \\quad (x > 1),\\\\[8pt]<\/p>\n

\\frac{d}{dx} (\\tanh^{-1} x)
\n&=\\frac{1}{\\sech^2 (\\tanh^{-1} x)} \\\\[4pt]
\n&=\\frac{1}{{1-\\tanh^2 (\\tanh^{-1} x)}} \\\\[4pt]
\n&=\\frac{1}{{1-x^2}} \\quad (\\lvert x \\rvert < 1),\\\\[8pt]\n\n\\frac{d}{dx} (\\coth^{-1} x )\n&=\\frac{1}{-\\csch^2 (\\coth^{-1} x)} \\\\[4pt]\n&=\\frac{1}{1-\\coth^2 (\\coth^{-1} x)} \\\\[4pt]\n&=\\frac{1}{1-x^2} \\quad (\\lvert x \\rvert > 1),\\\\[8pt]<\/p>\n

\\frac{d}{dx} (\\sech^{-1} x)
\n&=\\frac{d}{dx} \\left( \\cosh^{-1} \\frac{1}{x} \\right) \\\\[4pt]
\n&=\\frac{1}{\\sqrt{\\frac{1}{x^2}-1}} \\cdot \\left( -\\frac{1}{x^2}\\right) \\\\[4pt]
\n&= – \\frac{1}{x\\sqrt{1-x^2}} \\quad (0 < x < 1),\\\\[8pt]\n\n\\frac{d}{dx} (\\csch^{-1} x)\n&=\\frac{d}{dx} \\sinh^{-1} \\frac{1}{x} \\\\[4pt]\n&=\\frac{1}{\\sqrt{1+\\frac{1}{x^2}}} \\cdot \\left( -\\frac{1}{x^2}\\right) \\\\[4pt]\n&= - \\frac{1}{\\lvert x \\rvert \\sqrt{1+x^2}} \\quad (x \\ne 0).\n\n\\end{align}\\]\n\n\uc774\uac83\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2. (\uc5ed\uc30d\uace1\uc120\ud568\uc218\uc758 \ub3c4\ud568\uc218)<\/span><\/p>\n

\\[\\begin{align}
\n\\frac{d}{dx} (\\sinh^{-1} x ) &=\\frac{1}{\\sqrt{1+x^2}} &(&x\\in\\mathbb{R}),\\\\[4pt]
\n\\frac{d}{dx} (\\cosh^{-1} x ) &=\\frac{1}{\\sqrt{x^2 -1}} &(&x > 1),\\\\[4pt]
\n\\frac{d}{dx} (\\tanh^{-1} x) &=\\frac{1}{{1-x^2}} &(&\\lvert x \\rvert < 1),\\\\[4pt]\n\\frac{d}{dx} (\\coth^{-1} x ) &=\\frac{1}{1-x^2} &(&\\lvert x \\rvert > 1),\\\\[4pt]
\n\\frac{d}{dx} (\\sech^{-1} x) &= – \\frac{1}{x\\sqrt{1-x^2}} &(&0 < x < 1),\\\\[4pt]\n\\frac{d}{dx} (\\csch^{-1} x) &= - \\frac{1}{\\lvert x \\rvert \\sqrt{1+x^2}} &(&x \\ne 0).\n\\end{align}\\]\n<\/div>\n

<\/p>\n

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

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc30d\uace1\uc120\ud568\uc218\uc640 \uc5ed\uc30d\uace1\uc120\ud568\uc218\ub97c \uc815\uc758\ud558\uace0 \uc774 \ud568\uc218\ub4e4\uc758 \ub3c4\ud568\uc218\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \uc30d\uace1\uc120\ud568\uc218\uc758 \uc815\uc758 \\(\\mathbb{R}\\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\ub294 \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ud569\uc73c\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc989 \ud568\uc218 \\(f\\)\ub294 \\(f(x) = \\frac{f(x)-f(-x)}{2} + \\frac{f(x)+f(-x)}{2}\\) \ub85c\uc11c \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ub41c\ub2e4. \uc774\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc790\uc5f0\uc9c0\uc218\ud568\uc218\ub97c \uae30\ud568\uc218\uc640 \uc6b0\ud568\uc218\uc758 \ud569\uc73c\ub85c \ud45c\ud604\ud558\uba74 \\(e^x = \\frac{e^x – e^{-x}}{2} + \\frac{e^x + e^{-x}}{2}\\) \uc774\ub2e4. \uc774\ub54c \\(e^x\\)\uc758 \uae30\ud568\uc218 \ubd80\ubd84\uc744 \uc30d\uace1\uc120\uc0ac\uc778, \uc6b0\ud568\uc218 \ubd80\ubd84\uc744 \uc30d\uace1\uc120\ucf54\uc0ac\uc778\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc989 \uc30d\uace1\uc120\uc0ac\uc778(hyperbolic sine)\uc774\ub780 \\(\\sinh…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[236,84,189,233,234,235,237,185,231,232],"class_list":["post-1954","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-cosine","tag-derivative","tag-differentiation","tag-hyperbolic-function","tag-inverse-hyperbolic-function","tag-sine","tag-tangent","tag-185","tag-231","tag-232"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1954","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=1954"}],"version-history":[{"count":52,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1954\/revisions"}],"predecessor-version":[{"id":9158,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1954\/revisions\/9158"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1954"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1954"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}