Table of derivatives

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Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Mean value theorem | Vector calculus | Tensor calculus

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals

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.

Contents 1 Rules for differentiation of general functions 2 Derivatives of simple functions 3 Derivatives of exponential and logarithmic functions 4 Derivatives of trigonometric functions 5 Derivatives of hyperbolic functions

Rules for differentiation of general functions

<math>\left({cf}\right)' = cf'</math>

<math>\left({f + g}\right)' = f' + g'</math>

<math>\left({f - g}\right)' = f' - g'</math>

<math>\left({fg}\right)' = f'g + fg'</math>

<math>\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0</math>

<math>(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0</math>

<math>(f \circ g)' = (f' \circ g)g'</math>

<math>f' = (\ln f)'f, \qquad f > 0</math>

Derivatives of simple functions

<math>{d \over dx} c = 0</math>

<math>{d \over dx} x = 1</math>

<math>{d \over dx} cx = c</math>

<math>{d \over dx} |x| = {x \over |x|} = \sgn x,\qquad x \ne 0</math>

<math>{d \over dx} x^c = cx^{c-1} \qquad \mbox{where both } x^c \mbox{ and } cx^{c-1} \mbox { are defined}</math>

<math>{d \over dx} \left({1 \over x^c}\right) = {d \over dx} \left(x^{-c}\right) = -{c \over x^{c+1}}</math>

<math>{d \over dx} \left({1 \over x}\right) = {d \over dx} \left(x^{-1}\right) = -{1 \over x^2}</math>

<math>{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</math>

Derivatives of exponential and logarithmic functions

<math>{d \over dx} c^x = {c^x \ln c},\qquad c > 0</math>

<math>{d \over dx} e^x = e^x</math>

<math>{d \over dx} \log_c x = {1 \over x \ln c},\qquad c > 0, c \ne 1</math>

<math>{d \over dx} \ln x = {1 \over x}</math>

Derivatives of trigonometric functions

<math>{d \over dx} \sin x = \cos x</math>

<math>{d \over dx} \cos x = -\sin x</math>

<math>{d \over dx} \tan x = \sec^2 x</math>

<math>{d \over dx} \sec x = \tan x \sec x</math>

<math>{d \over dx} \cot x = -\csc^2 x</math>

<math>{d \over dx} \csc x = -\csc x \cot x</math>

<math>{d \over dx} \arcsin x = { 1 \over \sqrt{1 - x^2}}</math>

<math>{d \over dx} \arccos x = {-1 \over \sqrt{1 - x^2}}</math>

<math>{d \over dx} \arctan x = { 1 \over 1 + x^2}</math>

<math>{d \over dx} \arcsec x = { 1 \over |x|\sqrt{x^2 - 1}}</math>

<math>{d \over dx} \arccot x = {-1 \over 1 + x^2}</math>

<math>{d \over dx} \arccsc x = {-1 \over |x|\sqrt{x^2 - 1}}</math>

Derivatives of hyperbolic functions

<math>{d \over dx} \sinh x = \cosh x</math>

<math>{d \over dx} \cosh x = \sinh x</math>

<math>{d \over dx} \tanh x = \mbox{sech}^2 x</math>

<math>{d \over dx} \mbox{sech} x = - \tanh x \mbox{sech} x</math>

<math>{d \over dx} \mbox{coth} x = - \mbox{csch}^2 x</math>

<math>{d \over dx} \mbox{csch} x = - \mbox{coth} x \mbox{csch} x</math>

<math>{d \over dx} \mbox{arcsinh} x = { 1 \over \sqrt{x^2 + 1}}</math>

<math>{d \over dx} \mbox{arccosh} x = {-1 \over \sqrt{x^2 - 1}}</math>

<math>{d \over dx} \mbox{arctanh} x = { 1 \over 1 - x^2}</math>

<math>{d \over dx} \mbox{arcsech} x = { 1 \over x\sqrt{1 - x^2}}</math>

<math>{d \over dx} \mbox{arccoth} x = { 1 \over 1 - x^2}</math>

<math>{d \over dx} \mbox{arccsch} x = {-1 \over |x|\sqrt{1 + x^2}}</math>es:Tabla de derivadas

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