# 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 functions of x, and c is a constant. The set of real numbers is assumed. These formulas are sufficient to differentiate any elementary function.

 Contents

### Rules for differentiation of general functions

[itex]\left({cf}\right)' = cf'[itex]
[itex]\left({f + g}\right)' = f' + g'[itex]
[itex]\left({f - g}\right)' = f' - g'[itex]
[itex]\left({fg}\right)' = f'g + fg'[itex]
[itex]\left({f \over g}\right)' = {f'g - fg' \over g^2}[itex]
[itex](f^g)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0[itex]
[itex](f \circ g)' = (f' \circ g)g'[itex]

### Derivatives of simple functions

[itex]{d \over dx} c = 0[itex]
[itex]{d \over dx} x = 1[itex]
[itex]{d \over dx} |x| = {x \over |x|} = \sgn x,\qquad x \ne 0[itex]
[itex]{d \over dx} x^c = cx^{c-1}[itex]
[itex]{d \over dx} \sqrt{x} = {1 \over 2 \sqrt{x}}[itex]
[itex]{d \over dx} \left({1 \over x}\right) = -{1 \over x^2}[itex]

### Derivatives of exponential and logarithmic functions

[itex]{d \over dx} c^x = {c^x \ln c},\qquad c > 0[itex]
[itex]{d \over dx} e^x = e^x[itex]
[itex]{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1[itex]
[itex]{d \over dx} \ln x = {1 \over x}[itex]

### Derivatives of trigonometric functions

[itex]{d \over dx} \sin x = \cos x[itex]
[itex]{d \over dx} \cos x = -\sin x[itex]
[itex]{d \over dx} \tan x = \sec^2 x[itex]
[itex]{d \over dx} \sec x = \tan x \sec x[itex]
[itex]{d \over dx} \cot x = -\csc^2 x[itex]
[itex]{d \over dx} \csc x = -\cot x \csc x[itex]
[itex]{d \over dx} \arcsin x = { 1 \over \sqrt{1 - x^2}}[itex]
[itex]{d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}[itex]
[itex]{d \over dx} \arctan x = { 1 \over 1 + x^2}[itex]
[itex]{d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}[itex]
[itex]{d \over dx} \arccot x = {-1 \over 1 + x^2}[itex]
[itex]{d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}[itex]

### Derivatives of hyperbolic functions

[itex]{d \over dx} \sinh x = \cosh x[itex]
[itex]{d \over dx} \cosh x = \sinh x[itex]
[itex]{d \over dx} \tanh x = \mbox{sech}^2\,x[itex]
[itex]{d \over dx} \,\mbox{sech}\,x = -\tanh x\,\mbox{sech}\,x[itex]
[itex]{d \over dx} \,\mbox{coth}\,x = -\,\mbox{csch}^2\,x[itex]
[itex]{d \over dx} \,\mbox{csch}\,x = -\,\mbox{coth}\,x\,\mbox{csch}\,x[itex]
[itex]{d \over dx} \sinh^{-1} x = { 1 \over \sqrt{x^2 + 1}}[itex]
[itex]{d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}[itex]
[itex]{d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}[itex]
[itex]{d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}[itex]
[itex]{d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}[itex]
[itex]{d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}[itex]ro:Tabel de derivate

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