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 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) (2x)

14. Combine quotient rule and chain rule