List of differentiation identities: Information from Answers.com

Fundamental theorem
Limits of functions
Continuity
Mean value theorem

Differential calculus
Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Identities Rules: Power rule, Product rule, Quotient rule, Chain rule

Integral calculus
Integral Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order

Vector calculus
Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem

Multivariable calculus
Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian

It has been suggested that Techniques for differentiation be merged into this article or section. (Discuss)

It has been suggested that Differentiation rules be merged into this article or section. (Discuss)

The primary operation in differential calculus is finding a derivative. This table lists derivatives of many functions. In the following, f and g are differentiable functions, from the real numbers, and c is a real number. These formulas are sufficient to differentiate any elementary function.

1 General differentiation rules
2 Derivatives of simple functions
3 Derivatives of exponential and logarithmic functions
4 Derivatives of trigonometric functions
5 Derivatives of hyperbolic functions
6 Derivatives of special functions

General differentiation rules

Main article: Differentiation rules

Linearity: $\left({cf}\right)' = cf'$; $\left({f + g}\right)' = f' + g'$
Product rule: $\left({fg}\right)' = f'g + fg'$
Reciprocal rule: $\left(\frac{1}{f}\right)' = \frac{-f'}{f^2}, \qquad f \ne 0$
Quotient rule: $\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0$
Chain rule: $(f \circ g)' = (f' \circ g)g'$
Derivative of inverse function: $(f^{-1})' =\frac{1}{f' \circ f^{-1}}$

for any differentiable function f of a real argument and with real values, when the indicated compositions and inverses exist.

Generalized power rule: $(f^g)'=f^g \left( g'\ln f + \frac{g}{f} f' \right)$

Derivatives of simple functions

$c' = 0 \,$

$x' = 1 \,$

$(cx)' = c \,$

$|x|' = {x \over |x|} = \sgn x,\qquad x \ne 0$

$(x^c)' = cx^{c-1} \qquad \mbox{where both } x^c \mbox{ and } cx^{c-1} \mbox { are defined}$

$\left({1 \over x}\right)' = \left(x^{-1}\right)' = -x^{-2} = -{1 \over x^2}$

$\left({1 \over x^c}\right)' = \left(x^{-c}\right)' = -cx^{-(c+1)} = -{c \over x^{c+1}}$

$\left(\sqrt{x}\right)' = \left(x^{1\over 2}\right)' = {1 \over 2} x^{-{1\over 2}} = {1 \over 2 \sqrt{x}} , \qquad x > 0$

Derivatives of exponential and logarithmic functions

$\left(c^x\right)' = {c^x \ln c } ,\qquad c > 0$

note that the equation above is true for all c, but the derivative yields a complex number.

$\left(e^x\right)' = e^x$

$\left( \log_c x\right)' = {1 \over x \ln c} , \qquad c > 0, c \ne 1$

the equation above is also true for all c but yields a complex number.

$\left( \ln x\right)' = {1 \over x}, \qquad x \ne 0$

$\left( \ln |x|\right)' = {1 \over x}$

$\left( x^x \right)' = x^x(1+\ln x)$

The derivative of the natural logarithm with a generalised functional argument f(x) is

$\frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)}$

By applying the change-of-base rule, the derivative for other bases is

$\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.$

Derivatives of trigonometric functions

For more details on this topic, see Differentiation of trigonometric functions.

$(\sin x)' = \cos x \,$	$(\sin^{-1} x)' = { 1 \over \sqrt{1 - x^2}} \,$
$(\cos x)' = -\sin x \,$	$(\cos^{-1} x)' = {-1 \over \sqrt{1 - x^2}} \,$
$(\tan x)' = \sec^2 x = { 1 \over \cos^2 x} \,$	$(\tan^{-1} x)' = { 1 \over 1 + x^2} \,$
$(\sec x)' = \sec x \tan x \,$	$(\sec^{-1} x)' = { 1 \over \|x\|\sqrt{x^2 - 1}} \,$
$(\csc x)' = -\csc x \cot x \,$	$(\csc^{-1} x)' = {-1 \over \|x\|\sqrt{x^2 - 1}} \,$
$(\cot x)' = -\csc^2 x = { -1 \over \sin^2 x} \,$	$(\cot^{-1} x)' = {-1 \over 1 + x^2} \,$

Derivatives of hyperbolic functions

$( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}$	$(\operatorname{arcsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}$
$(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}$	$(\operatorname{arccosh}\,x)' = { 1 \over \sqrt{x^2 - 1}}$
$(\tanh x )'= \operatorname{sech}^2\,x$	$(\operatorname{arctanh}\,x)' = { 1 \over 1 - x^2}$
$(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x$	$(\operatorname{arcsech}\,x)' = {-1 \over x\sqrt{1 - x^2}}$
$(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x$	$(\operatorname{arccsch}\,x)' = {-1 \over x\sqrt{1 + x^2}}$
$(\operatorname{coth}\,x )' = -\,\operatorname{csch}^2\,x$	$(\operatorname{arccoth}\,x)' = { -1 \over x^2-1}$

Derivatives of special functions

Gamma function

$(\Gamma(x))' = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt$ $(\Gamma(x))' = \Gamma(x) \left(\sum_{n=1}^\infty \left(\ln\left(1 + \dfrac{1}{n}\right) - \dfrac{1}{x + n}\right) - \dfrac{1}{x}\right) = \Gamma(x) \psi(x)$

Riemann Zeta function

$(\zeta(x))' = -\sum_{n=1}^\infty \frac{\ln n}{n^x} = -\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots \!$

$(\zeta(x))' = -\sum_{p \text{ prime}} \frac{p^{-x} \ln p}{(1-p^{-x})^2}\prod_{q \text{ prime}, q \neq p} \frac{1}{1-q^{-x}} \!$

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	Techniques for differentiation
	Limits of integration
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List of differentiation identities

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