List of differentiation identities Information & List of differentiation identities Links at HealthHaven.com

Topics in Calculus
Differential calculus
	Derivative; Change of variables; Implicit differentiation; Taylor's theorem; Related rates; Identities; Rules:; Power rule, Product rule, Quotient rule, Chain rule
Integral calculus
	Integral; Lists of integrals; Improper integrals; Integration by:; parts, disks, cylindrical; shells, substitution,; trigonometric substitution,; partial fractions, changing order
Vector calculus
	Gradient; Divergence; Curl; Laplacian; Gradient theorem; Green's theorem; Stokes' theorem; Divergence theorem
Multivariable calculus
	Matrix calculus; Partial derivative; Multiple integral; Line integral; Surface integral; Volume integral; Jacobian

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.

[edit] General differentiation rules

Main article: Differentiation rules

Linearity: $\left({cf}\right)' = cf'$; $\left({f + g}\right)' = f' + g'$
Product rule: $\left({fg}\right)' = f'g + fg'$
Reciprocal rule: $\left(\frac{1}{f}\right)' = \frac{-f'}{f^2}, \qquad f \ne 0$
Quotient rule: $\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0$
Chain rule: $(f \circ g)' = (f' \circ g)g'$
Derivative of inverse function: $(f^{-1})' =\frac{1}{f' \circ f^{-1}}$

for any differentiable function f of a real argument and with real values, when the indicated compositions and inverses exist.

Generalized power rule: $(f^g)'=f^g \left( g'\ln f + \frac{g}{f} f' \right)$
Derivative of implicit function: If implicit function $y (x)$ is defined as $F (x, y (x)) = 0$; then; $y'_x=-\frac{F'_x}{F'_y}=-\frac{\partial F}{\partial x}/\frac{\partial F}{\partial y}$
Derivative of parametrically defined function: If a function $y (x)$ defined parametrically; $\begin{cases} x = f(t)\\ y = g(t) \end{cases}$; then; $y'_x=\frac{g'(f^{-1}(x))}{f'(f^{-1}(x))}$
Derivative of complex function.: For complex function $z (t) = F (u (t), v (t))$; the full derivative is; $z'=F'_u u'+F'_v v'=\frac{\partial F}{\partial u}u'+\frac{\partial F}{\partial v} v'\,$; If $z (x) = F (x, y (x))$; Then full derivative is; $z'=F'_x +F'_y y'=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} y'\,$

[edit] Derivatives of simple functions

$c'=0 \,$

$x' = 1 \,$

$(cx)' = c \,$

$|x|' = {x \over |x|} = \sgn x,\qquad x \ne 0$

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

$\left({1 \over x}\right)' = \left(x^{-1}\right)' = -x^{-2} = -{1 \over x^2}$

$\left({1 \over x^c}\right)' = \left(x^{-c}\right)' = -cx^{-(c+1)} = -{c \over x^{c+1}}$

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

[edit] Derivatives of exponential and logarithmic functions

$\left(c^x\right)' = {c^x \ln c } ,\qquad c > 0$

note that the equation above is true for all c, but the derivative yields a complex number.

$\left(e^x\right)' = e^x$

$\left( \log_c x\right)' = {1 \over x \ln c} , \qquad c > 0, c \ne 1$

the equation above is also true for all c but yields a complex number.

$\left( \ln x\right)' = {1 \over x}, \qquad x \ne 0$

$\left( \ln |x|\right)' = {1 \over x}$

$\left( x^x \right)' = x^x(1+\ln x)$

The derivative of the natural logarithm with a generalised functional argument f(x) is

$\frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)}$

By applying the change-of-base rule, the derivative for other bases is

$\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.$

[edit] Derivatives of trigonometric functions

For more details on this topic, see Differentiation of trigonometric functions.

$(\sin x)' = \cos x \,$	$(\sin^{-1} x)' = { 1 \over \sqrt{1 - x^2}} \,$
$(\cos x)' = -\sin x \,$	$(\cos^{-1} x)' = -{1 \over \sqrt{1 - x^2}} \,$
$(\tan x)' = \sec^2 x = { 1 \over \cos^2 x} \,$	$(\tan^{-1} x)' = { 1 \over 1 + x^2} \,$
$(\sec x)' = \sec x \tan x \,$	$(\sec^{-1} x)' = { 1 \over \|x\|\sqrt{x^2 - 1}} \,$
$(\csc x)' = -\csc x \cot x \,$	$(\csc^{-1} x)' = -{1 \over \|x\|\sqrt{x^2 - 1}} \,$
$(\cot x)' = -\csc^2 x = { -1 \over \sin^2 x} \,$	$(\cot^{-1} x)' = -{1 \over 1 + x^2} \,$

[edit] Derivatives of hyperbolic functions

$( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}$	$(\operatorname{arsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}$
$(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}$	$(\operatorname{arcosh}\,x)' = { 1 \over \sqrt{x^2 - 1}}$
$(\tanh x )'= \operatorname{sech}^2\,x$	$(\operatorname{artanh}\,x)' = { 1 \over 1 - x^2}$
$(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x$	$(\operatorname{arsech}\,x)' = -{1 \over x\sqrt{1 - x^2}}$
$(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x$	$(\operatorname{arcsch}\,x)' = -{1 \over x\sqrt{1 + x^2}}$
$(\operatorname{coth}\,x )' = -\,\operatorname{csch}^2\,x$	$(\operatorname{arcoth}\,x)' = -{ 1 \over x^2-1}$

[edit] Derivatives of special functions

Gamma function

$(\Gamma(x))' = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt$ $(\Gamma(x))' = \Gamma(x) \left(\sum_{n=1}^\infty \left(\ln\left(1 + \dfrac{1}{n}\right) - \dfrac{1}{x + n}\right) - \dfrac{1}{x}\right) = \Gamma(x) \psi(x)$

Riemann Zeta function

$(\zeta(x))' = -\sum_{n=1}^\infty \frac{\ln n}{n^x} = -\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots \!$

$(\zeta(x))' = -\sum_{p \text{ prime}} \frac{p^{-x} \ln p}{(1-p^{-x})^2}\prod_{q \text{ prime}, q \neq p} \frac{1}{1-q^{-x}} \!$

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