\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc0bc\uac01\ud568\uc218\uc640 \uc5ed\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n
<\/p>\n
\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc720\ub3c4\ud560 \ub54c\uc5d0\ub294 \uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148 \uacf5\uc2dd
\n\\[\\sin (x+h) = \\sin x \\cos h + \\cos x \\sin h\\]
\n\uc640 \uadf9\ud55c \uacf5\uc2dd
\n\\[\\lim_{h\\to 0}\\frac{\\cos h -1}{h} = 0 ,\\quad \\lim_{h\\to 0}\\frac{\\sin h}{h} =1\\]
\n\uc774 \uc0ac\uc6a9\ub41c\ub2e4. \uc774 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uc0ac\uc778 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\n\\frac{d}{dx} \\sin x
\n&= \\lim_{h\\to 0} \\frac{\\sin(x+h) – \\sin x}{h} \\\\[6pt]
\n&= \\lim_{h\\to 0} \\frac{(\\sin x \\cos h + \\cos x \\sin h) – \\sin x}{h} \\\\[6pt]
\n&= \\lim_{h\\to 0} \\frac{\\sin x ( \\cos h -1) + \\cos x \\sin h}{h} \\\\[6pt]
\n&= \\lim_{h\\to 0} \\left( \\sin x \\cdot \\frac{\\cos h -1}{h} \\right) + \\lim_{h\\to 0} \\left( \\cos x \\cdot \\frac{\\sin h}{h} \\right )\\\\[6pt]
\n&= \\sin x \\cdot \\lim_{h\\to 0}\\frac{\\cos h -1}{h} + \\cos x \\cdot \\lim_{h\\to 0} \\frac{\\sin h}{h} \\\\[8pt]
\n&= \\sin x \\cdot 0 + \\cos x \\cdot 1\\\\[8pt]
\n&= \\cos x.
\n\\end{align}\\]
\n\uc989
\n\\[\\frac{d}{dx} \\sin x = \\cos x \\]
\n\uc774\ub2e4. \ub610\ud55c
\n\\[\\cos x = \\sin \\left( x+ \\frac{\\pi}{2} \\right)\\]
\n\uc774\ubbc0\ub85c
\n\\[\\begin{align}
\n\\frac{d}{dx} \\cos x
\n&= \\frac{d}{dx} \\sin \\left( x+ \\frac{\\pi}{2} \\right) \\\\[6pt]
\n&= \\cos \\left( x + \\frac{\\pi}{2} \\right) \\\\[8pt]
\n&= – \\sin x
\n\\end{align}\\]
\n\uc989
\n\\[\\frac{d}{dx} \\cos x = -\\sin x\\]
\n\uc774\ub2e4. \ud55c\ud3b8
\n\\[\\begin{align}
\n\\tan x &= \\frac{\\sin x}{\\cos x},\\\\[6pt]
\n\\sec x &= \\frac{1}{\\cos x},\\\\[6pt]
\n\\csc x &= \\frac{1}{\\sin x},\\\\[6pt]
\n\\cot x &= \\frac{\\cos x}{\\sin x}
\n\\end{align}\\]
\n\uc774\ubbc0\ub85c
\n\\[\\begin{align}
\n\\frac{d}{dx} \\tan x
\n&= \\frac{d}{dx} \\frac{\\sin x}{\\cos x} \\\\[6pt]
\n&= \\frac{\\cos x \\cdot \\cos x – \\sin x \\cdot (-\\sin x)}{(\\cos x)^2} \\\\[6pt]
\n&= \\frac{1}{(\\cos x)^2} = \\sec^2 x ,\\\\[8pt]<\/p>\n
\\frac{d}{dx} \\sec x
\n&= \\frac{d}{dx} \\frac{1}{\\cos x} \\\\[6pt]
\n&= \\frac{-(-\\sin x)}{(\\cos x)^2} = \\sec x \\cdot \\tan x ,\\\\[8pt]<\/p>\n
\\frac{d}{dx} \\csc x
\n&= \\frac{d}{dx} \\frac{1}{\\sin x} \\\\[6pt]
\n&= \\frac{-\\cos x}{(\\sin x)^2} = -\\csc x \\cdot \\cot x ,\\\\[8pt]<\/p>\n
\\frac{d}{dx}\\cot x
\n&= \\frac{d}{dx} \\frac{\\cos x}{\\sin x}\\\\[6pt]
\n&= \\frac{-\\sin x \\cdot \\sin x – \\cos x \\cdot \\cos x}{(\\sin x)^2}\\\\[6pt]
\n&= -\\frac{1}{(\\sin x)^2} = -\\csc^2 x
\n\\end{align}\\]
\n\uc774\ub2e4. \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n<\/p>\n
\uc815\ub9ac 1. (\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84)<\/span><\/p>\n \\[\\begin{align*} <\/p>\n \uc0bc\uac01\ud568\uc218\ub294 \uc77c\ub300\uc77c \ud568\uc218\uac00 \uc544\ub2c8\uae30 \ub54c\ubb38\uc5d0 \uadf8 \uc5ed\ud568\uc218\uac00 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4. \uadf8\ub7ec\ub098 \uc0bc\uac01\ud568\uc218\uc758 \uc815\uc758\uc5ed\uc744 \ucd95\uc18c\ud558\uc5ec \uc77c\ub300\uc77c \ud568\uc218\uac00 \ub418\ub3c4\ub85d \ub9cc\ub4e6\uc73c\ub85c\uc368 \uadf8 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774\uc640 \uac19\uc740 \uad00\uc810\uc5d0\uc11c \uc0bc\uac01\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud558\uc790.<\/p>\n \\[\\begin{align} \uc815\uc758\uc5d0 \uc758\ud558\uc5ec \uc5ed\uc0bc\uac01\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc815\uc758\uc5ed\uc744 \uac00\uc9c4\ub2e4. \uc774\uc81c \uc774 \ud568\uc218\ub4e4\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud574 \ubcf4\uc790. \uba3c\uc800 \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc5ed\uc0ac\uc778\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4. \uc5ed\uc2dc\ucee8\ud2b8 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uacfc\uc815\uc740 \uc55e\uc758 \ub450 \ud568\uc218\uc5d0 \ube44\ud574 \uae4c\ub2e4\ub86d\ub2e4. \uc77c\ub2e8 \\(0\\)\uacfc \\(\\pi\\)\uc5d0\uc11c \ucf54\uc0ac\uc778\uc758 \ubbf8\ubd84\uacc4\uc218\ub294 \\(0\\)\uc774\ubbc0\ub85c \uc5ed\ud568\uc218\uc758 \ubbf8\ubd84\ubc95\uc5d0 \uc758\ud558\uc5ec \uc5ed\uc2dc\ucee8\ud2b8\ub294 \\(-1\\)\uacfc \\(1\\)\uc744 \uc81c\uc678\ud55c \uc810\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4. \uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc744 \uc774\uc6a9\ud558\uc5ec \uc5ed\uc2dc\ucee8\ud2b8\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc790. \ub2e4\ub978 \uc138 \uc5ed\uc0bc\uac01\ud568\uc218\uc758 \ub3c4\ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uad6c\ud55c\ub2e4. (\uc774 \uacf5\uc2dd\uc740 \ud55c \ubcc0\uc758 \uae38\uc774\uac00 \\(x\\)\uc778 \uc9c1\uac01\uc0bc\uac01\ud615\uc744 \uc774\uc6a9\ud558\uc5ec \uc99d\uba85\ud560 \uc218 \uc788\ub2e4.) \uc815\ub9ac 2. (\uc5ed\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84)<\/span><\/p>\n \\[\\begin{align} <\/p>\n","protected":false},"excerpt":{"rendered":" \uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \uc0bc\uac01\ud568\uc218\uc640 \uc5ed\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84 \uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84 \uacf5\uc2dd\uc744 \uc720\ub3c4\ud560 \ub54c\uc5d0\ub294 \uc0bc\uac01\ud568\uc218\uc758 \ub367\uc148 \uacf5\uc2dd \\(\\sin (x+h) = \\sin x \\cos h + \\cos x \\sin h\\) \uc640 \uadf9\ud55c \uacf5\uc2dd \\(\\lim_{h\\to 0}\\frac{\\cos h -1}{h} = 0 ,\\, \\lim_{h\\to 0}\\frac{\\sin h}{h} =1\\) \uc774 \uc0ac\uc6a9\ub41c\ub2e4. \uc774 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc5ec \uc0ac\uc778 \ud568\uc218\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4. \\(\\begin{align} \\frac{d}{dx} \\sin x &= \\lim_{h\\to 0} \\frac{\\sin(x+h) – \\sin…<\/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":[84,189,221,220,190,185,150,222],"class_list":["post-1946","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-derivative","tag-differentiation","tag-inverse-trigonometric-function","tag-trigonometric-function","tag-190","tag-185","tag-150","tag-222"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1946","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=1946"}],"version-history":[{"count":36,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1946\/revisions"}],"predecessor-version":[{"id":3044,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1946\/revisions\/3044"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1946"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1946"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1946"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\frac{d}{dx} \\sin x &= \\cos x , &\\frac{d}{dx} \\cos x &= -\\sin x ,\\\\[6pt]
\n\\frac{d}{dx} \\tan x &= \\sec^2 x ,& \\frac{d}{dx} \\cot x &= – \\csc^2 x, \\\\[6pt]
\n\\frac{d}{dx} \\sec x &= \\sec x \\,\\tan x ,&\\frac{d}{dx} \\csc x &= – \\csc x \\, \\cot x .
\n\\end{align*}\\]\n<\/p><\/div>\n\uc5ed\uc0bc\uac01\ud568\uc218\uc758 \ubbf8\ubd84<\/h3>\n
\ny = \\sin^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x=\\sin y \\,\\,\\,\\text{for}\\,\\,\\, y\\in \\left[ -\\frac{\\pi}{2} ,\\, \\frac{\\pi}{2}\\right],\\\\[6pt]
\ny = \\cos^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x=\\cos y \\,\\,\\,\\text{for}\\,\\,\\, y\\in \\left[ 0 ,\\, \\pi\\right],\\\\[6pt]
\ny = \\tan^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x=\\tan y \\,\\,\\,\\text{for}\\,\\,\\, y\\in \\left( -\\frac{\\pi}{2} ,\\, \\frac{\\pi}{2}\\right),\\\\[6pt]
\ny = \\cot^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x=\\cot y \\,\\,\\,\\text{for}\\,\\,\\, y\\in \\left( 0 ,\\, \\pi\\right),\\\\[6pt]
\ny = \\sec^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x=\\sec y \\,\\,\\,\\text{for}\\,\\,\\, y\\in \\left[ 0 ,\\, \\pi \\right] \\setminus \\left\\{\\frac{\\pi}{2}\\right\\},\\\\[6pt]
\ny = \\csc^{-1} x \\,\\, &\\Longleftrightarrow \\,\\, x=\\csc y \\,\\,\\,\\text{for}\\,\\,\\, y\\in \\left[ -\\frac{\\pi}{2} ,\\,\\frac{\\pi}{2} \\right] \\setminus \\left\\{ 0 \\right\\}.
\n\\end{align}\\]
\n\uc0bc\uac01\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub97c \ub098\ud0c0\ub0bc \ub54c \uc9c0\uc218\uc5d0 \\(-1\\)\uc744 \uc4f0\ub294 \ub300\uc2e0 \uc55e\uc5d0 \\(\\text{arc}\\)\ub97c \ubd99\uc774\uae30\ub3c4 \ud55c\ub2e4. \uc989 \uc704 \uc5ec\uc12f \uac1c\uc758 \uc5ed\uc0bc\uac01\ud568\uc218 \uc911 \uccab \uc138 \uac1c\ub294 \uac01\uac01
\n\\[\\arcsin x ,\\,\\, \\arccos x ,\\,\\, \\arctan x\\]
\n\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n
\n\\[\\begin{align}
\ny=\\sin^{-1}x \\,\\,\\,&\\text{for}\\,\\,\\, x\\in [-1,\\,1], \\\\[6pt]
\ny=\\cos^{-1}x \\,\\,\\,&\\text{for}\\,\\,\\, x\\in [-1,\\,1], \\\\[6pt]
\ny=\\tan^{-1}x \\,\\,\\,&\\text{for}\\,\\,\\, x\\in \\mathbb{R}, \\\\[6pt]
\ny=\\cot^{-1}x \\,\\,\\,&\\text{for}\\,\\,\\, x\\in \\mathbb{R}, \\\\[6pt]
\ny=\\sec^{-1}x \\,\\,\\,&\\text{for}\\,\\,\\, x\\in (-\\infty ,\\, -1] \\cup [1,\\, \\infty ), \\\\[6pt]
\ny=\\csc^{-1}x \\,\\,\\,&\\text{for}\\,\\,\\, x\\in (-\\infty ,\\, -1] \\cup [1,\\, \\infty ).
\n\\end{align}\\]
\n\uc0bc\uac01\ud568\uc218\uc758 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud558\uae30 \uc704\ud558\uc5ec \uc815\uc758\uc5ed\uc744 \ucd95\uc18c\ud558\ub294 \ubc29\ubc95\uc740 \uc704\uc640 \uac19\uc740 \ubc29\ubc95\uc774 \uc720\uc77c\ud55c \uac83\uc740 \uc544\ub2c8\ub2e4. \ucc45\uc5d0 \ub530\ub77c\uc11c \ub2e4\ub978 \uc815\uc758\uc5ed\uacfc \ub2e4\ub978 \uce58\uc5ed\uc744 \uc124\uc815\ud55c \ud6c4 \uc5ed\ud568\uc218\ub97c \uc815\uc758\ud558\uae30\ub3c4 \ud558\uba70, \ud604\uc2e4 \uc0c1\ud669\uc5d0\uc11c \uc751\uc6a9\ud560 \ub54c \uc0c1\ud669\uc5d0 \ub9de\uac8c \uc815\uc758\uc5ed\uacfc \uce58\uc5ed\uc744 \uc124\uc815\ud558\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n
\n\\[\\begin{align}
\n\\frac{d}{dx} \\sin^{-1} x
\n&= \\frac{1}{\\cos (\\sin^{-1} x )}\\\\[6pt]
\n&= \\frac{1}{\\sqrt{1-\\sin^2 (\\sin^{-1} x)}} \\\\[6pt]
\n&= \\frac{1}{\\sqrt{1-x^2}} \\,\\,\\,(\\lvert x \\rvert < 1).\n\\end{align}\\]\n\ub2e4\uc74c\uc73c\ub85c \uc5ed\ud0c4\uc820\ud2b8\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{align}\n\\frac{d}{dx}\\tan^{-1} x\n&= \\frac{1}{\\sec^2 (\\tan^{-1} x )} \\\\[6pt]\n&= \\frac{1}{1+\\tan^2 (\\tan^{-1} x )} \\\\[6pt]\n&= \\frac{1}{1+x^2}.\n\\end{align}\\]\n\uc704 \ub4f1\uc2dd\uc740 \ubaa8\ub4e0 \uc2e4\uc218 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \uc131\ub9bd\ud55c\ub2e4.<\/p>\n
\n\\[\\begin{align}
\ny&= \\sec^{-1} x,\\\\[8pt]
\n\\sec y &= x ,\\\\[6pt]
\n\\frac{d}{dx} \\sec y &= \\frac{d}{dx} x, \\\\[6pt]
\n\\sec y \\, \\tan y \\frac{dy}{dx} &= 1, \\\\[6pt]
\n\\frac{dy}{dx} &= \\frac{1}{\\sec y \\, \\tan y} .
\n\\end{align}\\]
\n\uc5ec\uae30\uc5d0
\n\\[\\sec y =x\\]
\n\uc640
\n\\[\\tan y = \\pm \\sqrt{\\sec^2 y -1} = \\pm \\sqrt{x^2 -1}\\]
\n\uc744 \ub300\uc785\ud558\uba74
\n\\[\\frac{dy}{dx} = \\pm \\frac{1}{x\\sqrt{x^2 -1}} \\,\\,\\,(\\lvert x \\rvert > 1)\\]
\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7f0\ub370 \\((0 ,\\,\\pi)\\)\uc5d0\uc11c \ucf54\uc0ac\uc778\uc758 \ub3c4\ud568\uc218\ub294 \ud56d\uc0c1 \uc591\uc218\uc774\ubbc0\ub85c \uadf8 \uc5ed\ud568\uc218\uc778 \uc2dc\ucee8\ud2b8\uc758 \ub3c4\ud568\uc218 \ub610\ud55c \uc591\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\frac{dy}{dx} = \\frac{1}{\\lvert x \\rvert \\sqrt{x^2 -1}}\\,\\,\\,(\\lvert x \\rvert > 1)\\]
\n\uc744 \uc5bb\ub294\ub2e4. \uc774\ub85c\uc368 \uc5ed\uc0ac\uc778, \uc5ed\uc2dc\ucee8\ud2b8, \uc5ed\ud0c4\uc820\ud2b8\uc758 \ub3c4\ud568\uc218\ub97c \uad6c\ud558\uc600\ub2e4.<\/p>\n
\n\\[\\begin{align}
\n\\cos^{-1} x = \\frac{\\pi}{2} – \\sin^{-1} x,\\\\[6pt]
\n\\cot^{-1} x = \\frac{\\pi}{2} – \\tan^{-1} x,\\\\[6pt]
\n\\csc^{-1} x = \\frac{\\pi}{2} – \\sec^{-1} x.
\n\\end{align}\\]
\n\uc989
\n\\[\\begin{align}
\n\\frac{d}{dx} \\cos^{-1} x &= \\frac{d}{dx} \\left( \\frac{\\pi}{2} – \\sin^{-1} x \\right) = – \\frac{1}{\\sqrt{1-x^2}} \\,\\,\\, (\\lvert x \\rvert < 1),\\\\[6pt]\n\\frac{d}{dx} \\cot^{-1} x &= \\frac{d}{dx} \\left( \\frac{\\pi}{2} - \\tan^{-1} x \\right) = - \\frac{1}{1+x^2} \\,\\,\\, ( x \\in \\mathbb{R}),\\\\[6pt]\n\\frac{d}{dx} \\csc^{-1} x &= \\frac{d}{dx} \\left( \\frac{\\pi}{2} - \\sec^{-1} x \\right) = - \\frac{1}{\\lvert x \\rvert \\sqrt{x^2 -1}} \\,\\,\\, (\\lvert x \\rvert > 1)
\n\\end{align}\\]
\n\uc774\ub2e4. \uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n<\/p>\n
\n\\frac{d}{dx} \\sin^{-1} x &= \\frac{1}{\\sqrt{1-x^2}} \\,\\,\\, (\\lvert x \\rvert < 1) ,\\\\[6pt]\n\\frac{d}{dx} \\cos^{-1} x &= -\\frac{1}{\\sqrt{1-x^2}} \\,\\,\\, (\\lvert x \\rvert < 1) ,\\\\[6pt]\n\\frac{d}{dx} \\tan^{-1} x &= \\frac{1}{1+x^2} \\,\\,\\, (x\\in\\mathbb{R}) ,\\\\[6pt]\n\\frac{d}{dx} \\cot^{-1} x &= -\\frac{1}{1+x^2} \\,\\,\\, (x\\in\\mathbb{R}) ,\\\\[6pt]\n\\frac{d}{dx} \\sec^{-1} x &= \\frac{1}{\\lvert x \\rvert \\sqrt{x^2 -1}} \\,\\,\\, (\\lvert x \\rvert > 1) ,\\\\[6pt]
\n\\frac{d}{dx} \\csc^{-1} x &= -\\frac{1}{\\lvert x \\rvert \\sqrt{x^2 -1}} \\,\\,\\, (\\lvert x \\rvert > 1) .
\n\\end{align}\\]\n<\/div>\n