List of differentiation identities

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Topics in calculus

Fundamental theorem
Limits of functions
Continuity
Vector calculus
Matrix calculus
Mean value theorem
Differentiation

Product rule
Quotient rule
Chain rule
Implicit differentiation
Taylor's theorem
Related rates
List of differentiation identities
Integration

Lists of integrals
Improper integrals
Integration by:
parts, disks, cylindrical
shells, substitution,
trigonometric substitution,
partial fractions, changing order
It has been suggested that Techniques for differentiation 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.

Contents

[edit] 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}
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.

[edit] Derivatives of simple functions

{d \over dx} c = 0
{d \over dx} x = 1
{d \over dx} (cx) = c
{d \over dx} |x| = {x \over |x|} = \sgn x,\qquad x \ne 0
{d \over dx} x^c = cx^{c-1} \qquad \mbox{where both } x^c \mbox{ and } cx^{c-1} \mbox { are defined}
{d \over dx} \left({1 \over x}\right) = {d \over dx} \left(x^{-1}\right) = -x^{-2} = -{1 \over x^2}
{d \over dx} \left({1 \over x^c}\right) = {d \over dx} \left(x^{-c}\right) = -cx^{-c-1} = -{c \over x^{c+1}}
{d \over dx} \sqrt{x} = {d \over dx} x^{1\over 2} = {1 \over 2} x^{-{1\over 2}} = {1 \over 2 \sqrt{x}}, \qquad x > 0

[edit] Derivatives of exponential and logarithmic functions

{d \over dx} c^x = {c^x \ln c },\qquad c > 0
{d \over dx} e^x = e^x
{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1
{d \over dx} \ln x = {1 \over x},\qquad x > 0
{d \over dx} \ln |x| = {1 \over x}
{d \over dx} x^x = x^x(1+\ln x)
(f^g)'=f^g \left( g'\ln f + \frac{g}{f} f' \right)

[edit] Derivatives of trigonometric functions

For more details on this topic, see Differentiation of trigonometric functions.
{d \over dx} \sin x = \cos x
{d \over dx} \cos x = -\sin x
{d \over dx} \tan x = \sec^2 x = { 1 \over \cos^2 x}
{d \over dx} \sec x = \sec x \tan x
{d \over dx} \csc x = -\csc x \cot x
{d \over dx} \cot x = -\csc^2 x = { -1 \over \sin^2 x}
{d \over dx} \arcsin x = { 1 \over \sqrt{1 - x^2}}
{d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}
{d \over dx} \arctan x = { 1 \over 1 + x^2}
{d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}
{d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}
{d \over dx} \arccot x = {-1 \over 1 + x^2}

[edit] Derivatives of hyperbolic functions

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

[edit] Derivatives of special functions

Gamma function

{d \over dx}\,\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt