The CHAIN RULE
(Derivatives of Composite Functions)
REVIEW – COMPOSITE FUNCTIONS
Given two functions, f(x) and g(x), the composite function,
f  g , is defined by f  g  f ( g( x )) .
outer
function
Example 
inner
function
If f(x) = x3 and g(x) = x + 1, determine:
a) f  g
b) g  f
c) f ( g(4))
The CHAIN RULE
If f and g are functions that have derivatives, and h( x )  f ( g( x )) , then
h '( x )  f '( g( x ))  g '( x ) .
(the derivative of the outer function multiplied by the derivative of the inner function)
Example 
If f ( x ) 
3
x2 ,
g( x )  x 2  x , and h( x )  f ( g( x )) , determine h'( x ) .
The power of a function
rule is a special case of
the chain rule!
Example 
Differentiate h(x) = (x2 + 3)4(4x – 5)3, expressing the answer in simplified
factored form.
Example 
Differentiate 𝑓 𝑥 =
2𝑥−1 6
𝑥+2
.
The CHAIN RULE in Leibniz Notation
If y is a function of u and u is a function of x, then y is a composite
dy dy du
dy
function and
and du exist.

 , provided
dx
dx du dx
du
Example 
a)
Determine
dy
in each of the following:
dx
y  u and u = 2x2 + 3
b)
y  u3  2u  1 and u  2 x
Example 
If y = u4 – 3u2 and u = x2 determine
dy
at x = –1 using the chain rule.
dx
Homework:
p.105–106 #7, 8, 10–14, 17b
Plus: Differentiate y = (x – 1)2(x + 3)5(2x – 5)3.