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.
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}<math>
- <math>(f^g)' = f^g\left(f'{g \over f} + g'\ln f\right),\qquad f > 0<math>
- <math>(f \circ g)' = (f' \circ g)g'<math>
Derivatives of simple functions
- <math>{d \over dx} c = 0<math>
- <math>{d \over dx} x = 1<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}<math>
- <math>{d \over dx} \sqrt{x} = {1 \over 2 \sqrt{x}}<math>
- <math>{d \over dx} \left({1 \over x}\right) = -{1 \over x^2}<math>
- <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>
- <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 = -\cot x \csc 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} \sinh^{-1} x = { 1 \over \sqrt{x^2 + 1}}<math>
- <math>{d \over dx} \cosh^{-1} x = {-1 \over \sqrt{x^2 - 1}}<math>
- <math>{d \over dx} \tanh^{-1} x = { 1 \over 1 - x^2}<math>
- <math>{d \over dx} \mbox{sech}^{-1}\,x = { 1 \over x\sqrt{1 - x^2}}<math>
- <math>{d \over dx} \mbox{coth}^{-1}\,x = {-1 \over 1 - x^2}<math>
- <math>{d \over dx} \mbox{csch}^{-1}\,x = {-1 \over |x|\sqrt{1 + x^2}}<math>ro:Tabel de derivate
