- List of differentiation identities
The primary operation in
differential calculus is finding aderivative . This table lists derivatives of many functions. In the following, "f" and "g" are differentiable functions, from thereal number s, 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 ofinverse 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 andlogarithm ic 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
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