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3.4
The Chain Rule
Copyright © Cengage Learning. All rights reserved.

2. The Chain Rule

3. The Chain Rule
The Chain Rule can be written either in the prime notation
(f  g)(x) = f (g(x))  g(x)
or, if y = f (u) and u = g(x), in Leibniz notation:
Equation 3 is easy to remember because if dy/du and du/dx were quotients, then we could cancel du.
Remember, however, that du has not been defined and du /dx should not be thought of as an actual quotient.

4. The Chain Rule
When applying the Chain Rule, it is helpful to think of the composite function f ◦ g as having two parts– an inner part and an outer part.
The derivative of y = f (u) is the derivative of the outer function (at the inner function u) times the derivative of the inner function.

5. Applying the General Power Rule
Find the derivative of f(x) = (3x – 2x2)3.
Solution:
Let u = 3x – 2x2.
Then f(x) = (3x – 2x2)3 = u3
and, by the General Power Rule, the derivative is

6. Trigonometric Functions and the Chain Rule
The “Chain Rule versions” of the derivatives of the six trigonometric functions are as follows.

7. The Chain Rule and Trigonometric Functions

8. Find the derivative (solutions to follow)

11. Outside/Inside method of chain rule
derivative of inside
derivative of outside times the inside
think of g(x) = u

12. derivative of outside times the inside
derivative of inside
inside
derivative of outside times the inside

13. derivative of outside times the inside
derivative of inside
derivative of outside times the inside

14. Outside/Inside method of chain rule
derivative of inside
derivative of outside times the inside

15. Example 3 Differentiate y = (x3 – 1)100. Solution:
Taking u = g(x) = x3 – 1 and n = 100 in , we have
= (x3 – 1)100
= 100(x3 – 1) (x3 – 1)
= 100(x3 – 1)99  3x2
= 300x2(x3 – 1)99

16. Chain Rule:
If is the composite of and , then:
example:
Find:

17. We could also do it this way:

18. Here is a faster way to find the derivative:
Differentiate the outside function...
…then the inside function

19. It looks like we need to use the chain rule again!
derivative of the
outside function
derivative of the
inside function

20. The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)