Differentiation rules

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 ) = a f ( x ) + b g ( x ) with respect to x is

h ′ ( x ) = a f ′ ( x ) + b g ′ ( x ) .

In Leibniz's notation this is written as:

d ( a f + b g ) d x = a d f d x + b d g d x .

Special cases include:

The constant factor rule

( a f ) ′ = a f ′

The sum rule

( 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 ) + f ( x ) g ′ ( x ) .

In Leibniz's notation this is written

d ( f g ) d x = d f d x g + f d g d x .

The 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 ) ) ⋅ g ′ ( x ) .

In Leibniz's notation this is correctly written as:

d d x h ( x ) = d d z f ( z ) | z = g ( x ) ⋅ d d x g ( x ) ,

often abridged to d h ( x ) d x = d f ( g ( x ) ) d g ( x ) ⋅ d g ( x ) d x . Focusing on the notion of maps, and the differential being a map D , this is written in a more concise way as:

[ D ( h ∘ g ) ] x = [ D h ] g ( x ) ⋅ [ D g ] x .

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 ′ = 1 f ′ ∘ g .

In Leibniz notation, this is written as

d x d y = 1 d y / d x .

The polynomial or elementary power rule

If f ( x ) = x r , for any real number r ≠ 0 then

f ′ ( x ) = r x 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

The derivative of h(x) = 1/f(x) for any (nonvanishing) function f is:

h ′ ( x ) = − f ′ ( x ) ( f ( x ) ) 2 .  

In Leibniz's notation, this is written

d ( 1 / f ) d x = − 1 f 2 d f d x .

The reciprocal rule can be derived from the quotient rule.

The quotient rule

If f and g are functions, then:

( f g ) ′ = f ′ g − g ′ f g 2 wherever g is nonzero.

This can be derived from product rule.

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 ) ′ = ( e g ln ⁡ f ) ′ = f g ( f ′ g f + g ′ ln ⁡ f ) ,

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

d d x ( c a x ) = c a x ln ⁡ c ⋅ a , c > 0

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

d d x ( e a x ) = a e a x d d x ( log c ⁡ x ) = 1 x ln ⁡ c , c > 0 , c ≠ 1

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

d d x ( ln ⁡ x ) = 1 x , x > 0. d d x ( ln ⁡ | x | ) = 1 x . d d x ( x x ) = x x ( 1 + ln ⁡ x ) . d d x ( f ( x ) g ( x ) ) = g ( x ) f ( x ) g ( x ) − 1 d f d x + f ( x ) g ( x ) ln ⁡ ( f ( x ) ) d g d x , if  f ( x ) > 0 ,  and if  d f d x  and  d g d x  exist. d d x ( f 1 ( x ) f 2 ( x ) ( . . . ) f n ( x ) ) = [ ∑ k = 1 n ∂ ∂ x k ( f 1 ( x 1 ) f 2 ( x 2 ) ( . . . ) f n ( x n ) ) ] | x 1 = x 2 = . . . = x n = x ,  if  f i < n ( x ) > 0  and  d f i d x  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 ) ′ = f ′ f wherever f is positive.

Derivatives of trigonometric functions

It is common to additionally define an inverse tangent function with two arguments, arctan ⁡ ( y , x ) . Its value lies in the range [ − π , π ] 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

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

F ( x ) = ∫ a ( x ) b ( x ) f ( x , t ) d t ,

where the functions f ( x , t ) and ∂ ∂ x f ( x , t ) are both continuous in both t and x in some region of the ( t , x ) plane, including a ( x ) ≤ t ≤ b ( x ) , x 0 ≤ x ≤ x 1 , and the functions a ( x ) and b ( x ) are both continuous and both have continuous derivatives for x 0 ≤ x ≤ x 1 . Then for x 0 ≤ x ≤ x 1 :

F ′ ( x ) = f ( x , b ( x ) ) b ′ ( x ) − f ( x , a ( x ) ) a ′ ( x ) + ∫ a ( x ) b ( x ) ∂ ∂ x f ( x , t ) d t .

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

d n d x n [ f ( g ( x ) ) ] = n ! ∑ { k m } f ( r ) ( g ( x ) ) ∏ m = 1 n 1 k m ! ( g ( m ) ( x ) ) k m

where r = ∑ m = 1 n − 1 k m and the set { k m } consists of all non-negative integer solutions of the Diophantine equation ∑ m = 1 n m k m = n .

General Leibniz rule

If f and g are n times differentiable, then

d n d x n [ f ( x ) g ( x ) ] = ∑ k = 0 n ( n k ) d n − k d x n − k f ( x ) d k d x k g ( x )