Chapter 7 Differential Calculus

Section 7.3 Calculation Rules

7.3.4 Composition of Functions

Finally, we investigate composition of functions (see Module 6, Section 6.6.3): what happens if a function u (the inner function) is substituted into another function v (the outer function)? In mathematics, such a composition is denoted by f:=vu with f(x)=(vu)(x):=v(u(x)). That is, first the value of a function u is determined depending on the variable x. The value u(x) calculated this way is then used as an argument of the function v. This results in the final function value v(u(x)).
Chain Rule 7.3.7
The derivative of the function f:=vu with f(x)=(vu)(x):=v(u(x)) can be calculated applying the chain rule:


Here, the expression v'(u(x)) is considered in such a way that v is a function of u and thus, the derivative is taken with respect to u; then v'(u) is evaluated for u=u(x).
The following phrase is a useful summary: the derivative of a composite function is the product of the outer derivative and the inner derivative.

This differentiation rule shall be illustrated by a few examples.
Example 7.3.8
Find the derivative of the function f: with f(x)=(3-2x )5 . To apply the chain rule, inner and outer functions must be identified. If we take the function u(x)=3-2x as the inner function u, then the outer function v is given by v(u)= u5 . With this, we have the required form v(u(x))=f(x).
Taking the derivative of the inner function u with respect to x results in u'(x)=-2. For the outer derivative, the function v is differentiated with respect to u, which results in v'(u)=5 u4 . Inserting these terms into the chain rule results in the derivative f' of the function f with

f'(x)=5(u(x ))4 ·(-2)=5(3-2x )4 ·(-2)=-10(3-2x )4 .

As a second example, let's calculate the derivative of g: with g(x)=e x3 . For the inner function u the assignment xu(x)= x3 and for the outer function v the assignment uv(u)=eu is appropriate. Taking the inner and the outer derivative results in u'(x)=3 x2 and v'(u)=eu . Inserting these terms into the chain rule results in the derivative of the function g:

g':,xg'(x)=eu(x) ·3 x2 =e x3 ·3 x2 =3 x2 e x3 .