MCV4U
2.5 The Derivatives of Composite Functions
Composition - the process of combining functions
Definition of a Composite Function
Given two functions f and g, the composite function (f o g) is defined by (f o g) = f(g(x))
Example #1: If f(x) = x2 and g(x) = x + 2, find each of the following values:
a) f(g(4))
b) g(f(2))
c) f(g(x))
d) g(f(x))
The chain rule is used to compute the derivative of the composite function h(x) = f(g(x)) in
terms of the derivatives of f and g.
The Chain Rule
In other words…
"the derivative of a composite function is the product of the derivative of the outer function
evaluated at the inner function and the derivative of the inner function."
The Chain Rule Leibniz Notation
If y is function of u, and u is a function of x or if y = f (u) and u = g(x) and they are differentiable
functions, then
If we interpret derivatives as rates of change, the chain rule states that if y is a
function of x through the intermediate variable u, then the rate of change of y
with respect to x is equal to the product of the rate of change of y with respect to
u and the rate of change of u with respect to x.
Use the power of a function rule when the outer function is a power of the function of the form
[g(x)]n or y = un.
Power of a Function Rule
1. Differentiate:
i) y = (4x – 3)5
ii) f(x) = (3x2 – 2x + 4)5
iii) g(x) = (x2 + x)3/2
MCV4U
iv)
vi)
2.5 The Derivatives of Composite Functions
6
√2
4
v)
5
3
√
vii)
2. Find the equation of the tangent to the graph of
at the point(2, 1)
Assigned work: page 105 - 106 #1, 2, 4 - 14, 16, 19