The Chain Rule

1. The Chain Rule • Working on the Chain Rule

2. Review of Derivative Rules • Using Limits:

3. Power Rule • If f(x) =

4. Product Rule

5. Quotient Rule

6. Why use the chain rule? • The previous rules work well to take derivatives of functions such as • How do you best find a derivative of an equation such as

7. The Chain Rule • The chain rule is used to calculate derivatives of composite functions, such as f(g(x)). • Ex: Let f(x)= and • Therefore, f(g(x))= • Obviously, it would be difficult to expand the above function. The best way to calculate the derivative is by use of the chain rule.

8. Chain Rule (cont) • The derivative of a composite function, f(g)x)), is found by multiplying the derivative of f(g(x)) by the derivative of g(x). • Or, f’(g(x))(g’(x)) • In our example, , we obtain • This is the general power rule of the chain rule

9. Other applications of the chain rule • To find f’(x) when f(x)=sin , f’(x)=(cos )(2x) • To find f’(x) when f(x)= rewrite the equation as • Then, use the general power rule of the chain rule to obtain

10. Trig and the Chain Rule • Let f(x)=sin u. f’(x)=(cos u)u’ • Ex: f(x)=sin2x, f’(x)=cos2x(2)=2cos2x • Find the following derivatives: • A. f(x)=cos(x-1) • B. f(x)=cos(2x) • C. f(x)=sin( )

11. A. f(x) = cos(x-1) f’(x) = -sin(x-1) • B. f(x) = cos(2x) f’(x) = -2sin(2x) • C. f(x) = sin(2 ) f’(x) = 4xcos(2 )

12. Combining Chain Rule • Let f(x)=sin(2x)cos(2x). Find f’(x)

13. Combine product rule and chain rule (2x) (2x) • Let h(x)=sin(2x)cos(2x). Find h’(x) • From product rule, d/dx f(x)g(x)= • f’(x)g(x) + f(x)g’(x) • From above, if f(x)=sin(2x) and g(x)=cos(2x), then f’(x)=2cos(2x) and g’(x)=-2sin(2x) • Therefore, h’(x)=(2cos(2x))(cos(2x)) + (sin(2x))(-2sin(2x)) = (2x) (2x)

14. Combine quotient rule and chain rule