# Differentiation rules

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

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

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

In Leibniz's notation this is written as:

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

Special cases include:

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

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

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 of a 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 as:

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

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

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

### The inverse function rule

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 $f(x) = x^n$, for any integer n 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.
• 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 chain rule and the power 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 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,

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

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

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

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

$\frac{d}{dx}\left(e^x\right) = e^x$
$\frac{d}{dx}\left( \log_c x\right) = {1 \over x \ln c} , \qquad c > 0, c \ne 1$

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

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

### 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

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

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 $\Gamma'(x) = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt$ $= \Gamma(x) \left(\sum_{n=1}^\infty \left(\ln\left(1 + \dfrac{1}{n}\right) - \dfrac{1}{x + n}\right) - \dfrac{1}{x}\right) = \Gamma(x) \psi(x)$
 Riemann Zeta function $\zeta'(x) = -\sum_{n=1}^\infty \frac{\ln n}{n^x} = -\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots \!$ $= -\sum_{p \text{ prime}} \frac{p^{-x} \ln p}{(1-p^{-x})^2}\prod_{q \text{ prime}, q \neq p} \frac{1}{1-q^{-x}} \!$

## Derivatives of integrals

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

If f and g are n times differentiable, then

$\frac{d^n}{d x^n} [f(g(x))]= n! \sum_{\{k_m\}}^{} f^{(r)}(g(x)) \prod_{m=1}^n \frac{1}{k_m!} \left(g^{(m)}(x) \right)^{k_m}$

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 $\sum_{m=1}^{n} m k_m = n$.

### General Leibniz rule

Main article: General Leibniz rule

If f and g are n times differentiable, then

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

## 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