# Differentiation rules

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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

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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:

$(af)'=af'\,$ $(f+g)'=f'+g'\,$ • The subtraction rule
$(f-g)'=f'-g'.\,$ ### The product rule

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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

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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

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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

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

{{#invoke:main|main}} 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

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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

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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.

${\frac {d}{dx}}\left(e^{x}\right)=e^{x}$ ${\frac {d}{dx}}\left(\log _{c}x\right)={1 \over x\ln c},\qquad c>0,c\neq 1$ the equation above is also true for all c but yields a complex number if c<0.

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

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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

## Derivatives of integrals

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

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

{{#invoke:main|main}} If f and g are n times differentiable, then

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

{{#invoke:main|main}} If f and g are n times differentiable, then

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