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Differentiation rules

Fundamental theorem
Limits of functions
Continuity
Mean value theorem

Differential calculus
Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities: 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

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

Elementary rules of differentiation

Unless otherwise stated, all functions will be functions from R to R, although more generally, the formulae below make sense wherever they are well defined.

Differentiation is linear

For any functions f and g and any real numbers a and b.

$(af + bg)' = af' + bg'.\,$

In other words, the derivative of the function h(x) = a f(x) + b g(x) with respect to x is

$h'(x) = a f'(x) + b g'(x).\,$

In Leibniz's notation this is written

$\frac{d(af+bg)}{dx} = a\frac{df}{dx} +b\frac{dg}{dx}.$

Special cases include:

The constant multiple rule

$(af)' = a\,f' \,$

The sum rule

$(f + g)' = f' + g'\,$

The subtraction rule

$(f - g)' = f' - g'.\,$

The product or Leibniz rule

Main article: Product rule

For any of the functions f and g,

$(fg)' = f' g + f g'.\,$

In other words, the derivative of the function h(x) = f(x) g(x) with respect to x is

$h'(x) = f'(x) g(x) + f(x) g'(x).\,$

In Leibniz's notation this is written

$\frac{d(fg)}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}.$

The chain rule

This is a rule for computing the derivative of a function of a function, i.e., of the composite $f\circ g$ of two functions f and g:

$(f \circ g)' = (f' \circ g)g'.\,$

In other words, the derivative of the function h(x) = f(g(x)) with respect to x is

$h'(x) = f'(g(x)) g'(x).\,$

In Leibniz's notation this is written (suggestively) as:

$\frac{df}{dx} = \frac{df}{dg} \frac{dg}{dx}.\,$

The polynomial or elementary power rule

If f(x) = xⁿ, for some natural number n (including zero) then

$f'(x) = nx^{n-1}.\,$

Special cases include:

Constant rule: if f is the constant function f(x) = c, for any number c, then for all x

$f'(x) = 0.\,$

The derivative of a linear function is constant: if f(x) = ax (or more generally, in view of the constant rule, if f(x)=ax+b ), then

$f'(x) = a.\,$

Combining this rule with the linearity of the derivative permits the computation of the derivative of any polynomial.

The reciprocal rule

not

For any (nonvanishing) function f, the derivative of the function 1/f (equal at x to 1/f(x)) is

$-\frac{f'}{f^2}.\,$

In other words, the derivative of h(x) = 1/f(x) is

$h'(x) = -\frac{f'(x)}{[f(x)]^2}.\$

In Leibniz's notation, this is written

$\frac{d(1/f)}{dx} = -\frac{1}{f^2}\frac{df}{dx}.\,$

The inverse function rule

This should not be confused with the reciprocal rule: the reciprocal 1/x of a nonzero real number x is its inverse with respect to multiplication, whereas the inverse of a function is its inverse with respect to function composition.

If the function f has an inverse g = f⁻¹ (so that g(f(x)) = x and f(g(y)) = y) then

$g' = \frac{1}{f'\circ f^{-1}}.\,$

In Leibniz notation, this is written (suggestively) as

$\frac{dx}{dy} = \frac{1}{dy/dx}.$

Further rules of differentiation

The quotient rule

Main article: Quotient rule

If f and g are functions, then:

$\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\quad$ wherever g is nonzero.

This can be derived from reciprocal rule and the product rule. Conversely (using the constant rule) the reciprocal rule is the special case f(x) = 1.

Generalized power rule

Main article: Power rules

The elementary power rule generalizes considerably. First, if x is positive, it holds when n is any real number. The reciprocal rule is then the special case n = -1 (although care must then be taken to avoid confusion with the inverse rule).

The most general power rule is the functional power rule: for any functions f and g,

$(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\quad$

wherever both sides are well defined.

Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):

$(\ln f)'= \frac{f'}{f} \quad$ wherever f is positive.

Derivatives of exponential and logarithmic functions

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

note that the equation above is true for all c, but the derivative for c < 0 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.

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

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

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

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

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

Derivatives of trigonometric functions

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

Derivatives of hyperbolic functions

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

Derivatives of special functions

Nth Derivatives

The following formulae can be obtained empirically by repeated differentiation and taking notice of patterns; either by hand or computed by a CAS (Computer Algebra System).^[1] Below y is the dependent variable, x is the independent variable, real number constants are A, B, N, r, real integers are n and j, F(x) is a continuously differentiable function (the n^th derivative exists), and i is the imaginary unit $\sqrt{-1} \!$ .

Function	n^th Derivative
$y = F(G(x)) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = n! \displaystyle\sum_{\{k_m\}}^{} \dfrac{\mathrm{d}^r}{\mathrm{d} z^r} F(z)\|_{z=G(x)} \displaystyle\prod_{m=1}^n \dfrac{1}{k_m!} \left(\dfrac{1}{m!} \dfrac{\mathrm{d}^m}{\mathrm{d} x^m} G(x) \right)^{k_m} \!$ where $r = \displaystyle\sum_{m=1}^{n-1} k_m \!$ and the set $\{k_m\} \!$ consists of all non-negative integer solutions of the Diophantine equation $\displaystyle\sum_{m=1}^{n} m k_m = n \!$ See: Faà di Bruno's formula, Expansions for nearly Gaussian distributions by S. Blinnikov and R. Moessner ^[2]
$y = F(x)G(x) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \displaystyle\sum_{k=0}^{n} \displaystyle\binom{n}{k} \dfrac{\mathrm{d}^{n-k}}{\mathrm{d} x^{n-k}} F(x) \dfrac{\mathrm{d}^k}{\mathrm{d} x^k} G(x) \!$ See: General Leibniz rule
$y = x^N \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \displaystyle\prod_{r=1}^n (N-r+1)x^{N-n} \!$
$y = [F(x)]^r \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = r \displaystyle\binom{n-r}{n} \displaystyle\sum_{j=0}^{n} \dfrac{(-1)^j}{r-j}{\displaystyle\binom{n}{j} [F(x)]^{r-j} \dfrac{\mathrm{d}^n}{\mathrm{d} x^n} [F(x)]^j} \!$
$y = B^{Ax} \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n B^{Ax} \left ( \ln{B} \right )^n \!$
For the case of $B = \exp(1) = e \!$ (the exponential function), the above reduces to: $y = e^{Ax} \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n e^{Ax} \!$
$y = \ln[F(x)] \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \delta_n \ln[F(x)] + \displaystyle\sum_{j=1}^n \dfrac{(-1)^{j-1}}{j} \binom{n}{j} \dfrac{1}{[F(x)]^j} \dfrac{\mathrm{d}^n}{\mathrm{d} x^n} [F(x)]^j \!$ where $\delta_n = \begin{cases} 1 & n=0 \\ 0 & n \neq 0 \\ \end{cases} \!$ is the Kronecker delta.
$y = \sin(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \sin \left ( A x + B + \frac{n \pi}{2} \right ) \!$ Expanding this by the sine addition formula yields a more clear form to use: $\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \left [ \cos \left (\dfrac{n \pi}{2} \right ) \sin \left ( A x + B \right ) + \sin \left ( \dfrac{n \pi}{2} \right ) \cos \left ( A x + B \right ) \right ] \!$
$y = \cos(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \cos \left ( A x + B + \frac{n \pi}{2} \right ) \!$ Expanding by the cosine addition formula: $\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \left [ \cos \left (\dfrac{n \pi}{2} \right ) \cos \left ( A x + B \right ) - \sin \left ( \dfrac{n \pi}{2} \right ) \sin \left ( A x + B \right ) \right ] \!$
$y = \sinh(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = (-i A^n) \sinh \left ( Ax + B + \dfrac{ i n \pi}{2} \right ) \!$
$y = \cosh(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = (\pm i A^n) \cosh \left ( Ax + B \mp \dfrac{ i n \pi}{2} \right ) \!$