Table of Derivatives

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Section A.3 Table of Derivatives

Throughout this table,

a

and

b

are constants, independent of

x .

$F (x)$	$F^{'} (x) = \frac{d F}{d x}$
$a f (x) + b g (x)$	$a f^{'} (x) + b g^{'} (x)$
$f (x) + g (x)$	$f^{'} (x) + g^{'} (x)$
$f (x) - g (x)$	$f^{'} (x) - g^{'} (x)$
$a f (x)$	$a f^{'} (x)$
$f (x) g (x)$	$f^{'} (x) g (x) + f (x) g^{'} (x)$
$f (x) g (x) h (x)$	$f^{'} (x) g (x) h (x) + f (x) g^{'} (x) h (x) + f (x) g (x) h^{'} (x)$
$\frac{f (x)}{g (x)}$	$\frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{g (x)^{2}}$
$\frac{1}{g (x)}$	$- \frac{g^{'} (x)}{g (x)^{2}}$
$f (g (x))$	$f^{'} (g (x)) g^{'} (x)$

$F (x)$	$F^{'} (x) = \frac{d F}{d x}$
$a$	$0$
$x^{a}$	$a x^{a - 1}$
$g (x)^{a}$	$a g (x)^{a - 1} g^{'} (x)$
$\sin x$	$\cos x$
$\sin g (x)$	$g^{'} (x) \cos g (x)$
$\cos x$	$- \sin x$
$\cos g (x)$	$- g^{'} (x) \sin g (x)$
$\tan x$	$\sec^{2} x$
$\csc x$	$- \csc x \cot x$
$\sec x$	$\sec x \tan x$
$\cot x$	$- \csc^{2} x$
$e^{x}$	$e^{x}$
$e^{g (x)}$	$g^{'} (x) e^{g (x)}$
$a^{x}$	$(\ln a) a^{x}$

$F (x)$	$F^{'} (x) = \frac{d F}{d x}$
$\ln x$	$\frac{1}{x}$
$\ln g (x)$	$\frac{g^{'} (x)}{g (x)}$
$\log_{a} x$	$\frac{1}{x \ln a}$
$\arcsin x$	$\frac{1}{\sqrt{1 - x^{2}}}$
$\arcsin g (x)$	$\frac{g^{'} (x)}{\sqrt{1 - g (x)^{2}}}$
$\arccos x$	$- \frac{1}{\sqrt{1 - x^{2}}}$
$\arctan x$	$\frac{1}{1 + x^{2}}$
$\arctan g (x)$	$\frac{g^{'} (x)}{1 + g (x)^{2}}$
$arccsc x$	$- \frac{1}{\| x \| \sqrt{x^{2} - 1}}$
$arcsec x$	$\frac{1}{\| x \| \sqrt{x^{2} - 1}}$
$arccot x$	$- \frac{1}{1 + x^{2}}$