導関数と原始関数の表 - 知識メモ - atwiki（アットウィキ）

微分前	微分後
$x^{\alpha}$	$\alpha x^{\alpha - 1}$
$\{f(x)\}^{\alpha}$	$\alpha \{f(x)\}^{\alpha - 1} \cdot f'(x)$
$\{f(x)\}^{1/2}$	$\frac{f'(x)}{2 \sqrt{f(x)}}$
$\log{ \mid f(x) \mid }$	$\frac{f'(x)}{f(x)}$
$\sin{x}$	$\cos{x}$
$\cos{x}$	$- \sin{x}$
$\tan{x}$	$\frac{1}{\cos^2{x}} = \sec^2{x}$
$\frac{1}{\tan{x}}$	$- \frac{1}{\sin^2{x}} = - cosec^2{x}$
$e^x$	$e^x$
$e^{f(x)}$	$f'(x) \cdot e^{f(x)}$
$a^x$	$a^x \log{a}$
$a^{f(x)}$	$f'(x) \cdot a^{f(x)} \log{a}$
$x^x$	$x^x (1 + \log{x})$
$f(x)g(x)$	$f&#039;(x)g(x) + f(x)g&#039;(x)$
$\frac{f(x)}{g(x)}$	$\frac{f'(x)g(x) - f(x)g'(x)}{ \{g(x)\}^2 }$
$Arcsin \ x$	$\frac{1}{\sqrt{1-x^2}}$
$Arccos \ x$	$- \frac{1}{\sqrt{1-x^2}}$
$Arctan \ x$	$\frac{1}{1+x^2}$
$Arcsin \left( \frac{x}{a} \right)$	$\frac{1}{\sqrt{a^2-x^2}} \ \ (a > 0)$
$\log{\mid x + \sqrt{x^2+a} \mid}$	$\frac{1}{\sqrt{x^2+a}} \ \ (a \neq 0)$
$\frac{1}{2} \left\{ x \sqrt{a^2-x^2} + a^2 \ Arcsin \left( \frac{x}{a} \right) \right\}$	$\sqrt{a^2-x^2} \ \ (a > 0)$
$\frac{1}{2} \left( x \sqrt{x^2+a} + a \log{\mid x + \sqrt{x^2+a} \mid} \right)$	$\sqrt{x^2+a} \ \ (a \neq 0)$
$\frac{1}{a} Arctan \left( \frac{x}{a} \right)$	$\frac{1}{x^2+a^2} \ \ (a \neq 0)$
$\frac{1}{2a} \log{\mid \frac{x-a}{x+a} \mid}$	$\frac{1}{x^2-a^2} \ \ (a \neq 0)$
微分前	微分後