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Table of derivatives: differentiation rules and common derivatives

Every derivative a first calculus course needs, on one page. The function is on the left, its derivative on the right.

Differentiation rules

Function	Derivative
$(c f)^{'}$	$c f^{'}$
$(f \pm g)^{'}$	$f^{'} \pm g^{'}$
$(f g)^{'}$	$f^{'} g + f g^{'}$
$(\frac{f}{g})^{'}$	$\frac{f ^{'} g - f g ^{'}}{g ^{2}}$
$(f (g (x)))^{'}$	$f^{'} (g (x)) \cdot g^{'} (x)$

Powers, exponentials and logarithms

Function	Derivative
$c$	$0$
$x^{n}$	$n x^{n - 1}$
$x$	$\frac{1}{2 x}$
$e^{x}$	$e^{x}$
$a^{x}$	$a^{x} ln a$
$ln x$	$\frac{1}{x}$
$lo g_{a} x$	$\frac{1}{x ln a}$

Trigonometric, inverse and hyperbolic

Function	Derivative
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$\frac{1}{cos ^{2} x}$
$cot x$	$- \frac{1}{sin ^{2} x}$
$arcsin x$	$\frac{1}{1 - x ^{2}}$
$arccos x$	$- \frac{1}{1 - x ^{2}}$
$arctan x$	$\frac{1}{1 + x ^{2}}$
$sinh x$	$cosh x$
$cosh x$	$sinh x$

Notes

The chain rule works from the outside in: the derivative of the outer function, evaluated at the inner one, times the derivative of the inner one.

The power rule holds for any real exponent, negative or fractional. For example √x = x^(1/2).