List of differentiation identities

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.

General differentiation rules

;Linearity:$left(\{cf\}\; ight)\text{'}\; =\; cf\text{'}$:$left(\{f\; +\; g\}\; ight)\text{'}\; =\; f\text{'}\; +\; g\text{'}$;Product rule:$left(\{fg\}\; ight)\text{'}\; =\; f\text{'}g\; +\; fg\text{'}$;Reciprocal rule:$left(frac\{1\}\{f\}\; ight)\text{'}\; =\; frac\{-f\text{'}\}\{f^2\}$;Quotient rule:$left(\{f\; over\; g\}\; ight)\text{'}\; =\; \{f\text{'}g\; -\; fg\text{'}\; over\; g^2\},\; qquad\; g\; e\; 0$;Chain rule:$(f\; circ\; g)\text{'}\; =\; (f\text{'}\; circ\; g)g\text{'}$;Derivative of inverse function:$(f^\{-1\})\text{'}\; =frac\{1\}\{f\text{'}\; circ\; f^\{-1$for any differentiable function "f" of a real argument and with real values, when the indicated compositions and inverses exist.;Generalized power rule:$(f^g)\text{'}=f^g\; left(\; g\text{'}ln\; f\; +\; frac\{g\}\{f\}\; f\text{'}\; ight)$

= Derivatives of simple functions =

: $c\text{'}\; =\; 0\; ,$

: $x\text{'}\; =\; 1\; ,$

: $(cx)\text{'}\; =\; c\; ,$

: $|x|\text{'}\; =\; \{x\; over\; |x\; =\; sgn\; x,qquad\; x\; e\; 0$

: $(x^c)\text{'}\; =\; cx^\{c-1\}\; qquad\; mbox\{where\; both\; \}\; x^c\; mbox\{\; and\; \}\; cx^\{c-1\}\; mbox\; \{\; are\; defined\}$

: $left(\{1\; over\; x\}\; ight)\text{'}\; =\; left(x^\{-1\}\; ight)\text{'}\; =\; -x^\{-2\}\; =\; -\{1\; over\; x^2\}$

: $left(\{1\; over\; x^c\}\; ight)\text{'}\; =\; left(x^\{-c\}\; ight)\text{'}\; =\; -cx^\{-c-1\}\; =\; -\{c\; over\; x^\{c+1$

: $left(sqrt\{x\}\; ight)\text{'}\; =\; left(x^\{1over\; 2\}\; ight)\text{'}\; =\; \{1\; over\; 2\}\; x^\{-\{1over\; 2\; =\; \{1\; over\; 2\; sqrt\{x,\; qquad\; x\; >\; 0$

= Derivatives of exponential and logarithmic functions =

:$left(c^x\; ight)\text{'}\; =\; \{c^x\; ln\; c\; \},qquad\; c\; >\; 0$

:$left(e^x\; ight)\text{'}\; =\; e^x$

:$left(\; log\_c\; x\; ight)\text{'}\; =\; \{1\; over\; x\; ln\; c\}\; qquad,\; c\; >\; 0,\; c\; e\; 1$

:$left(\; ln\; x\; ight)\text{'}\; =\; \{1\; over\; x\}\; qquad,\; x\; >\; 0$

:$left(\; ln\; |x|\; ight)\text{'}\; =\; \{1\; over\; x\}$

:$left(\; x^x\; ight)\text{'}\; =\; x^x(1+ln\; x)$

Derivatives of trigonometric functions

:$(sin\; x)\text{'}\; =\; cos\; x\; ,$

:$(cos\; x)\text{'}\; =\; -sin\; x\; ,$

:$(\; an\; x)\text{'}\; =\; sec^2\; x\; =\; \{\; 1\; over\; cos^2\; x\}\; ,$

:$(sec\; x)\text{'}\; =\; sec\; x\; an\; x\; ,$

:$(csc\; x)\text{'}\; =\; -csc\; x\; cot\; x\; ,$

:$(cot\; x)\text{'}\; =\; -csc^2\; x\; =\; \{\; -1\; over\; sin^2\; x\}\; ,$:$(arcsin\; x)\text{'}\; =\; \{\; 1\; over\; sqrt\{1\; -\; x^2\; ,$

:$(arccos\; x)\text{'}\; =\; \{-1\; over\; sqrt\{1\; -\; x^2\; ,$

:$(arctan\; x)\text{'}\; =\; \{\; 1\; over\; 1\; +\; x^2\}\; ,$

:$(arcsec\; x)\text{'}\; =\; \{\; 1\; over\; |x|sqrt\{x^2\; -\; 1\; ,$

:$(arccsc\; x)\text{'}\; =\; \{-1\; over\; |x|sqrt\{x^2\; -\; 1\; ,$

:$(arccot\; x)\text{'}\; =\; \{-1\; over\; 1\; +\; x^2\}\; ,$col-end

= Derivatives of hyperbolic functions =

:$(\; sinh\; x\; )\text{'}=\; cosh\; x\; =\; frac\{e^x\; +\; e^\{-x\{2\}$

:$(\; cosh\; x\; )\text{'}=\; sinh\; x\; =\; frac\{e^x\; -\; e^\{-x\{2\}$

:$(\; anh\; x\; )\text{'}=\; operatorname\{sech\}^2,x$

:$(\; operatorname\{sech\},x)\text{'}\; =\; -\; anh\; x,operatorname\{sech\},x$

:$(\; operatorname\{csch\},x)\text{'}\; =\; -,operatorname\{coth\},x,operatorname\{csch\},x$

:$(operatorname\{coth\},x\; )\text{'}\; =\; -,operatorname\{csch\}^2,x$:$(operatorname\{arcsinh\},x)\text{'}\; =\; \{\; 1\; over\; sqrt\{x^2\; +\; 1$

:$(operatorname\{arccosh\},x)\text{'}\; =\; \{\; 1\; over\; sqrt\{x^2\; -\; 1$

:$(operatorname\{arctanh\},x)\text{'}\; =\; \{\; 1\; over\; 1\; -\; x^2\}$

:$(operatorname\{arcsech\},x)\text{'}\; =\; \{-1\; over\; xsqrt\{1\; -\; x^2$

:$(operatorname\{arccsch\},x)\text{'}\; =\; \{-1\; over\; xsqrt\{1\; +\; x^2$

:$(operatorname\{arccoth\},x)\text{'}\; =\; \{\; 1\; over\; 1\; -\; x^2\}$

Derivatives of special functions

Gamma function:$(Gamma(x))\text{'}\; =\; int\_0^infty\; t^\{x-1\}\; e^\{-t\}\; ln\; t,dt$