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# Differentiation rules

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 Title: Differentiation rules Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### 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—including complex numbers (C).

### 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

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

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

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

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

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

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

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)