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.

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

[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

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

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List of differentiation identities

From Wikipedia, the free encyclopedia

Contents

[edit] General differentiation rules

[edit] Derivatives of simple functions

[edit] Derivatives of exponential and logarithmic functions

[edit] Derivatives of trigonometric functions

[edit] Derivatives of hyperbolic functions

[edit] Derivatives of special functions

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