微分公式

基本公式
$$ \frac{d}{dx}x^n = nx^{n-1} $$ $$ \frac{d}{dx}e^x = e^x $$
指数・対数
$$ \frac{d}{dx}a^x = a^x \log a $$ $$ \frac{d}{dx}\log|x| = \frac{1}{x} $$ $$ \frac{d}{dx}\log _a|x| = \frac{1}{x\log a} $$
三角関数
$$ \frac{d}{dx}{\sin}x = {\cos}x $$ $$ \frac{d}{dx}{\cos}x = -{\sin}x $$ $$ \frac{d}{dx}{\tan}x = \frac{1}{\cos ^2 x} = \sec ^2 x $$
逆三角関数
$$ \frac{d}{dx}{\sin ^{-1}}x = \frac{1}{\sqrt{1-x^2}} $$ $$ \frac{d}{dx}{\cos ^{-1}}x = - \frac{1}{\sqrt{1-x^2}} $$ $$ \frac{d}{dx}{\tan ^{-1}}x = - \frac{1}{1 + x^2} $$
双曲線関数
$$ \frac{d}{dx}{\sinh}x = {\cosh}x $$ $$ \frac{d}{dx}{\cosh}x = -{\sinh}x $$ $$ \frac{d}{dx}{\tanh}x = \sec h ^2 x $$
逆双曲線関数
$$ \frac{d}{dx}{\sinh ^{-1}}x = \frac{d}{dx}\log(x + \sqrt{x^2 + 1}) = \frac{1}{\sqrt{x^2 + 1}} $$ $$ \frac{d}{dx}{\cosh ^{-1}}x = \frac{d}{dx}\log(x \pm \sqrt{x^2 - 1}) = \frac{d}{dx}\pm\log(x + \sqrt{x^2 - 1}) = \pm \frac{1}{\sqrt{x^2 - 1}}    (x > 1) $$ $$ \frac{d}{dx}{\tanh ^{-1}}x = \frac{d}{dx} \frac{1}{2} \log \frac{1+x}{1-x} = \frac{1}{1-x^2}    (|x| < 1) $$