Table Of Derivatives

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 Number s, and ''c'' is a real number. These formulas are sufficient to differentiate any Elementary Function .

Rules for differentiation of general functions

;Constant multiple rule
:

\left({cf}
ight)' = cf'

;Linearity
:

\left({f + g}
ight)' = f' + g'

\left({f - g}
ight)' = f' - g'

;Product rule
:

\left({fg}
ight)' = f'g + fg'

;Quotient rule
:

\left({f \over g}
ight)' = {f'g - fg' \over g^2}, \qquad g 
e 0

;Functional power rule
:

(f^g)' = \left(e^{g\ln f}
ight)' = f^g\left(f'{g \over f} + g'\ln f
ight),\qquad f > 0

;Chain rule
:

(f \circ g)' = (f' \circ g)g'

;Logarithm rule
:

(\ln f)' = rac{f'}{f}, \qquad f 
e 0

Derivatives of simple functions

:

{d \over dx} c = 0

{d \over dx} x = 1

{d \over dx} cx = c

{d \over dx} \left({1 \over x}
ight) = {d \over dx} \left(x^{-1}
ight) = {d \over dx} \left(-x^{-2}
ight) = -{1 \over x^2}

{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

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 
e 1

{d \over dx} \ln x = {1 \over x}

Derivatives of Trigonometric functions

:

{d \over dx} \sin x = \cos x

{d \over dx} \cos x = -\sin x

{d \over dx} 	an x = \sec^2 x

{d \over dx} \sec x = 	an x \sec x

{d \over dx} \cot x = -\csc^2 x

{d \over dx} \csc x = -\csc x \cot 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} \mbox{arccosh} x = { 1 \over \sqrt{x^2 - 1}}

{d \over dx} \mbox{arctanh} x = { 1 \over 1 - x^2}

{d \over dx} \mbox{arcsech} x = { 1 \over x\sqrt{1 - x^2}}

{d \over dx} \mbox{arccoth} x = { 1 \over 1 - x^2}

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Table Of Derivatives