1. The Chain Rule Find the composition of two functions.
OBJECTIVE
Find the composition of two functions.
Differentiate using the Extended Power Rule or the Chain Rule.

2. 1.7 The Chain Rule DEFINITION:
The composed function , the composition of f and g, is defined as

3. 1.7 The Chain Rule Example 1: For and find and
p. 172 works example 5 like example 5 was posed with parts (a) and (b) (but it wasn’t).

4. 1.7 The Chain Rule
Example 2: For and
find and

5. 1.7 The Chain Rule Quick Check 1 For the functions in Example 2, find:
b.)

6. 1.7 The Chain Rule Suppose you are asked to differentiate the function
The differentiation formulas you learned in the
previous sections of this chapter do not enable you
to calculate F’(x).

7. 1.7 The Chain Rule
Observe that F is a composite function. In fact, if we let and let u = g(x) = x2 + 1, then we can write y = F(x) = f (g(x)).
That is, F = f ◦ g.

8. 1.7 The Chain Rule We know how to differentiate both f and g.
So, it would be useful to have a rule that shows us how to find the derivative of F = f ◦ g in terms of the derivatives of f and g.

9. 1.7 The Chain Rule It turns out that the derivative of the composite
function f ◦ g is the product of the derivatives of f and g.
This fact is one of the most important of the differentiation rules. It is called the Chain Rule.

10. It seems plausible if we interpret derivatives as rates of change.
1.7 The Chain Rule
It seems plausible if we interpret derivatives as rates of change.
Regard:
du/dx as the rate of change of u with respect to x
dy/du as the rate of change of y with respect to u
dy/dx as the rate of change of y with respect to x

11. 1.7 The Chain Rule
If u changes twice as fast as x and y changes three times as fast as u, it seems reasonable that y changes six times as fast as x.
So, we expect that:

12. 1.7 The Chain Rule
The Chain Rule can be written either in the prime notation
(Equation 1)
or, if y = f(u) and u = g(x), in Leibniz notation:
(Equation 2)
Derivative of “outside” times derivative of “inside”

13. Find F '(x) if 1.7 The Chain Rule
One way of solving this is by using Equation 1.
At the beginning of this section, we expressed F
as F(x) = (f ◦ g))(x) = f(g(x)) where
and g(x) = x2 + 1.

14. 1.7 The Chain Rule
Since
we have

15. We can also solve by using Equation 2.
1.7 The Chain Rule
We can also solve by using Equation 2.
If we let and u = x2 + 1, then:

16. In using the Chain Rule, we work from the outside to the inside.
Equation 2 states that we differentiate the outer
function f [at the inner function g(x)] and then
we multiply by the derivative of the inner function.

17. THEOREM 7: The Extended Power Rule
1.7 The Chain Rule
THEOREM 7: The Extended Power Rule
Suppose that g(x) is a differentiable function
of x. Then, for any real number k,

18. 1.7 The Chain Rule
Example 3: Differentiate

19. 1.7 The Chain Rule Example 4: Differentiate
Combine Product Rule and Extended Power Rule

20. 1.7 The Chain Rule Quick Check 2 Differentiate:
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1.7 The Chain Rule
Quick Check 2
Differentiate:
We will combine both the quotient rule and the chain rule:

21. 1.7 The Chain Rule
Example 5: For and
find and
and

22. 1.7 The Chain Rule Section Summary
The Extended Power Rule tells us that if then
The composition of with is written and is defined as
In general,

23. Section Summary Concluded
1.7 The Chain Rule
Section Summary Concluded
The Chain Rule is used to differentiate a composition of functions. If
Then