List of differentiation identities

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.

General differentiation rules

;Linearity:$$left({cf} ight)' = cf':$$left({f + g} ight)' = f' + g';Product rule:$$left({fg} ight)' = f'g + fg';Reciprocal rule:$$left(frac{1}{f} ight)' = frac{-f'}{f^2};Quotient rule:$$left({f over g} ight)' = {f'g - fg' over g^2}, qquad g e 0;Chain rule:$$(f circ g)' = (f' circ g)g';Derivative of inverse function:$$(f^{-1})' =frac{1}{f' circ f^{-1for any differentiable function "f" of a real argument and with real values, when the indicated compositions and inverses exist.;Generalized power rule:$$(f^g)'=f^g left( g'ln f + frac{g}{f} f' ight)

= Derivatives of simple functions =

: $$c' = 0 ,

: $$x' = 1 ,

: $$(cx)' = c ,

: $$|x|' = {x over |x = sgn x,qquad x e 0

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

: $$left({1 over x} ight)' = left(x^{-1} ight)' = -x^{-2} = -{1 over x^2}

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

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

= Derivatives of exponential and logarithmic functions =

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

:$$left(e^x ight)' = e^x

:$$left( log_c x ight)' = {1 over x ln c} qquad, c > 0, c e 1

:$$left( ln x ight)' = {1 over x} qquad, x > 0

:$$left( ln |x| ight)' = {1 over x}

:$$left( x^x ight)' = x^x(1+ln x)

Derivatives of trigonometric functions

:$$ (sin x)' = cos x ,

:$$ (cos x)' = -sin x ,

:$$ ( an x)' = sec^2 x = { 1 over cos^2 x} ,

:$$ (sec x)' = sec x an x ,

:$$ (csc x)' = -csc x cot x ,

:$$ (cot x)' = -csc^2 x = { -1 over sin^2 x} ,:$$ (arcsin x)' = { 1 over sqrt{1 - x^2 ,

:$$ (arccos x)' = {-1 over sqrt{1 - x^2 ,

:$$ (arctan x)' = { 1 over 1 + x^2} ,

:$$ (arcsec x)' = { 1 over |x|sqrt{x^2 - 1 ,

:$$ (arccsc x)' = {-1 over |x|sqrt{x^2 - 1 ,

:$$ (arccot x)' = {-1 over 1 + x^2} ,col-end

= Derivatives of hyperbolic functions =

:$$( sinh x )'= cosh x = frac{e^x + e^{-x{2}

:$$( cosh x )'= sinh x = frac{e^x - e^{-x{2}

:$$( anh x )'= operatorname{sech}^2,x

:$$( operatorname{sech},x)' = - anh x,operatorname{sech},x

:$$( operatorname{csch},x)' = -,operatorname{coth},x,operatorname{csch},x

:$$(operatorname{coth},x )' = -,operatorname{csch}^2,x:$$(operatorname{arcsinh},x)' = { 1 over sqrt{x^2 + 1

:$$(operatorname{arccosh},x)' = { 1 over sqrt{x^2 - 1

:$$(operatorname{arctanh},x)' = { 1 over 1 - x^2}

:$$(operatorname{arcsech},x)' = {-1 over xsqrt{1 - x^2

:$$(operatorname{arccsch},x)' = {-1 over xsqrt{1 + x^2

:$$(operatorname{arccoth},x)' = { 1 over 1 - x^2}

Derivatives of special functions

Gamma function:$$(Gamma(x))' = int_0^infty t^{x-1} e^{-t} ln t,dt