Differentiation rules

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Specialized

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 are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined^[1]^[2]—including complex numbers (C).^[3]

Differentiation is linear

Main article: Linearity of differentiation

For any functions $f$ and $g$ and any real numbers $a$ and $b$ the derivative of the function $h(x)=af(x)+bg(x)$ with respect to $x$ is

h'(x)=af'(x)+bg'(x).\,

In Leibniz's notation this is written as:

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

Special cases include:

The constant factor rule

(af)'=af'\,

The sum rule

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

The subtraction rule

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

The product rule

Main article: Product rule

For the functions f and g, 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)+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

Main article: Chain rule

The derivative of the function $h(x)=f(g(x))$ with respect to $x$ is

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

In Leibniz's notation this is correctly written as:

{\frac {d}{dx}}h(x)={\frac {d}{dz}}f(z)|_{z=g(x)}\cdot {\frac {d}{dx}}g(x),\,

often abridged to $\quad {\frac {dh(x)}{dx}}={\frac {df(g(x))}{dg(x)}}\cdot {\frac {dg(x)}{dx}}.\,$ Focusing on the notion of maps, and the differential being a map ${\text{D}}$ , this is written in a more concise way as:

[{\text{D}}(h\circ g)]_{x}=[{\text{D}}h]_{g(x)}\cdot [{\text{D}}g]_{x}\,.\,

The inverse function rule

Main article: inverse functions and differentiation

If the function f has an inverse function g, meaning that g(f(x)) = x and f(g(y)) = y, then

g'={\frac {1}{f'\circ g}}.

In Leibniz notation, this is written as

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

Power laws, polynomials, quotients, and reciprocals

The polynomial or elementary power rule

Main article: Power rule

If $f(x) = x^r$ , for any real number $r\neq 0$ then

f'(x)=rx^{r-1}.\,

Special cases include:

If f(x) = x, then f′(x) = 1. This special case may be generalized to:
The derivative of an affine function is constant: if f(x) = ax + b, then f′(x) = a.

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

The reciprocal rule

Main article: Reciprocal rule

The derivative of h(x) = 1/f(x) for any (nonvanishing) function f 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 reciprocal rule can be derived from the quotient rule.

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 product rule.

Generalized power rule

Main article: Power rule

The elementary power rule generalizes considerably. 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.

Special cases:

If f(x) = x^a, f′(x) = ax^{a − 1} when a is any non-zero real number and x is positive.
The reciprocal rule may be derived as the special case where g(x) = −1.

Derivatives of exponential and logarithmic functions

{\frac {d}{dx}}\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.

{\frac {d}{dx}}\left(e^{ax}\right)=ae^{ax}

{\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1

the equation above is also true for all c but yields a complex number if c<0.

{\frac {d}{dx}}\left(\ln x\right)={1 \over x},\qquad x>0.

{\frac {d}{dx}}\left(\ln |x|\right)={1 \over x}.

{\frac {d}{dx}}\left(x^{x}\right)=x^{x}(1+\ln x).

{\frac {d}{dx}}\left(f(x)^{g(x)}\right)=g(x)f(x)^{g(x)-1}{\frac {df}{dx}}+f(x)^{g(x)}\ln {(f(x))}{\frac {dg}{dx}},\qquad {\text{if }}f(x)>0,{\text{ and if }}{\frac {df}{dx}}{\text{ and }}{\frac {dg}{dx}}{\text{ exist.}}

{\frac {d}{dx}}\left(f_{1}(x)^{f_{2}(x)^{\left(...\right)^{f_{n}(x)}}}\right)=\left[\sum \limits _{k=1}^{n}{\frac {\partial }{\partial x_{k}}}\left(f_{1}(x_{1})^{f_{2}(x_{2})^{\left(...\right)^{f_{n}(x_{n})}}}\right)\right]{\biggr \vert }_{x_{1}=x_{2}=...=x_{n}=x},{\text{ if }}f_{i<n}(x)>0{\text{ and }}

{\frac {df_{i}}{dx}}{\text{ exists. }}

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 trigonometric functions

Main article: Differentiation of trigonometric functions

$(\sin x)'=\cos x\,$	$(\arcsin x)'={1 \over {\sqrt {1-x^{2}}}}\,$
$(\cos x)'=-\sin x\,$	$(\arccos x)'=-{1 \over {\sqrt {1-x^{2}}}}\,$
$(\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,$	$(\arctan x)'={1 \over 1+x^{2}}\,$
$(\sec x)'=\sec x\tan x\,$	$(\operatorname {arcsec} x)'={1 \over \|x\|{\sqrt {x^{2}-1}}}\,$
$(\csc x)'=-\csc x\cot x\,$	$(\operatorname {arccsc} x)'=-{1 \over \|x\|{\sqrt {x^{2}-1}}}\,$
$(\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,$	$(\operatorname {arccot} x)'=-{1 \over 1+x^{2}}\,$

It is common to additionally define an inverse tangent function with two arguments, $\arctan(y,x)$ . Its value lies in the range $[-\pi ,\pi ]$ and reflects the quadrant of the point $(x,y)$ . For the first and fourth quadrant (i.e. $x>0$ ) one has $\arctan(y,x>0)=\arctan(y/x)$ . Its partial derivatives are

{\frac {\partial \arctan(y,x)}{\partial y}}={\frac {x}{x^{2}+y^{2}}}

, and

{\frac {\partial \arctan(y,x)}{\partial x}}={\frac {-y}{x^{2}+y^{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

Gamma function $\quad \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,dt$

\Gamma '(x)=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt

\,=\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)

with $\psi (x)$ being the digamma function, expressed by the parenthesized expression to the right of $\Gamma(x)$ in the line above.

Riemann Zeta function $\quad \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}$: $\zeta '(x)=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x}}}=-{\frac {\ln 2}{2^{x}}}-{\frac {\ln 3}{3^{x}}}-{\frac {\ln 4}{4^{x}}}-\cdots \!$

\,=-\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}}}\!

Derivatives of integrals

Main article: Differentiation under the integral sign

Suppose that it is required to differentiate with respect to x the function

F(x)=\int _{a(x)}^{b(x)}f(x,t)\,dt,

where the functions $f(x,t)\,$ and ${\frac {\partial }{\partial x}}\,f(x,t)\,$ are both continuous in both $t\,$ and $x\,$ in some region of the $(t,x)\,$ plane, including $a(x)\leq t\leq b(x),$ $x_{0}\leq x\leq x_{1}\,$ , and the functions $a(x)\,$ and $b(x)\,$ are both continuous and both have continuous derivatives for $x_{0}\leq x\leq x_{1}\,$ . Then for $\,x_{0}\leq x\leq x_{1}\,\,$ :

F'(x)=f(x,b(x))\,b'(x)-f(x,a(x))\,a'(x)+\int _{a(x)}^{b(x)}{\frac {\partial }{\partial x}}\,f(x,t)\;dt\,.

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.

Derivatives to nth order

Some rules exist for computing the nth derivative of functions, where n is a positive integer. These include:

Faà di Bruno's formula

Main article: Faà di Bruno's formula

If f and g are n times differentiable, then

{\frac {d^{n}}{dx^{n}}}[f(g(x))]=n!\sum _{\{k_{m}\}}^{}f^{(r)}(g(x))\prod _{m=1}^{n}{\frac {1}{k_{m}!}}\left(g^{(m)}(x)\right)^{k_{m}}

where $r=\sum _{m=1}^{n-1}k_{m}$ and the set $\{k_{m}\}$ consists of all non-negative integer solutions of the Diophantine equation $\sum _{m=1}^{n}mk_{m}=n$ .

General Leibniz rule

Main article: General Leibniz rule