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

${\displaystyle h'(x)=af'(x)+bg'(x).\,}$

In Leibniz's notation this is written as:

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

Special cases include:

${\displaystyle (af)'=af'\,}$
${\displaystyle (f+g)'=f'+g'\,}$
• The subtraction rule
${\displaystyle (f-g)'=f'-g'.\,}$

### 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

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

In Leibniz's notation this is written

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

### The chain rule

 Main article: Chain rule

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

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

In Leibniz's notation this is written as:

${\displaystyle {\frac {dh}{dx))={\frac {df(g(x))}{dg(x))){\frac {dg(x)}{dx)).\,}$

However, by relaxing the interpretation of h as a function, this is often simply written

${\displaystyle {\frac {dh}{dx))={\frac {dh}{dg)){\frac {dg}{dx)).\,}$

### 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

${\displaystyle g'={\frac {1}{f'\circ g)).}$

In Leibniz notation, this is written as

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

## Power laws, polynomials, quotients, and reciprocals

### The polynomial or elementary power rule

 Main article: Power rule

If ${\displaystyle f(x)=x^{n))$, for any integer n then

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

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.
• 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:

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

In Leibniz's notation, this is written

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

The reciprocal rule can be derived from the chain rule and the power rule.

### The quotient rule

 Main article: Quotient rule

If f and g are functions, then:

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

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

### 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,

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

wherever both sides are well defined.

Special cases:

• If f(x) = xa, f′(x) = axa − 1 when a is any 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

${\displaystyle {\frac {d}{dx))\left(c^{ax}\right)={c^{ax}\ln c\cdot a},\qquad c>0}$

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

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

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

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

### Logarithmic derivatives

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

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

## Derivatives of trigonometric functions

 Main article: Differentiation of trigonometric functions
 ${\displaystyle (\sin x)'=\cos x\,}$ ${\displaystyle (\arcsin x)'={1 \over {\sqrt {1-x^{2))))\,}$ ${\displaystyle (\cos x)'=-\sin x\,}$ ${\displaystyle (\arccos x)'=-{1 \over {\sqrt {1-x^{2))))\,}$ ${\displaystyle (\tan x)'=\sec ^{2}x={1 \over \cos ^{2}x}=1+\tan ^{2}x\,}$ ${\displaystyle (\arctan x)'={1 \over 1+x^{2))\,}$ ${\displaystyle (\sec x)'=\sec x\tan x\,}$ ${\displaystyle (\operatorname {arcsec} x)'={1 \over |x|{\sqrt {x^{2}-1))}\,}$ ${\displaystyle (\csc x)'=-\csc x\cot x\,}$ ${\displaystyle (\operatorname {arccsc} x)'=-{1 \over |x|{\sqrt {x^{2}-1))}\,}$ ${\displaystyle (\cot x)'=-\csc ^{2}x={-1 \over \sin ^{2}x}=-(1+\cot ^{2}x)\,}$ ${\displaystyle (\operatorname {arccot} x)'=-{1 \over 1+x^{2))\,}$

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

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

## Derivatives of hyperbolic functions

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

## Derivatives of special functions

 Gamma function ${\displaystyle \Gamma '(x)=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt}$ ${\displaystyle =\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n))\right)-{\dfrac {1}{x+n))\right)-{\dfrac {1}{x))\right)=\Gamma (x)\psi (x)}$
 Riemann Zeta function ${\displaystyle \zeta '(x)=-\sum _{n=1}^{\infty }{\frac {\ln n}{n^{x))}=-{\frac {\ln 2}{2^{x))}-{\frac {\ln 3}{3^{x))}-{\frac {\ln 4}{4^{x))}-\cdots \!}$ ${\displaystyle =-\sum _{p{\text{ prime))}{\frac {p^{-x}\ln p}{(1-p^{-x})^{2))}\prod _{q{\text{ prime)),q\neq p}{\frac {1}{1-q^{-x))}\!}$

## Derivatives of integrals

 Main article: Differentiation under the integral sign

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

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

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

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

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

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

### General Leibniz rule

 Main article: General Leibniz rule

If f and g are n times differentiable, then

${\displaystyle {\frac {d^{n)){dx^{n))}[f(x)g(x)]=\sum _{k=0}^{n}{\binom {n}{k)){\frac {d^{n-k)){dx^{n-k))}f(x){\frac {d^{k)){dx^{k))}g(x)}$

## References

1. ^ Calculus (5th edition), F. Ayres, E. Mendelson, Schuam's Outline Series, 2009, ISBN 978-0-07-150861-2.
2. ^ Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schuam's Outline Series, 2010, ISBN 978-0-07-162366-7.
3. ^ Complex Variables, M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3