1. Homework Homework Assignment #15 Read Section 3.7 Page 170, Exercises: 1 – 49 (EOO), 43 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

2. Homework, Page 170 Find an equation of the tangent line at the point indicated. 1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

3. Homework, Page 170 Find the derivative of each function. 5. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

4. Homework, Page 170 Find the derivative of each function. 9. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

5. Homework, Page 170 Find the derivative of each function. 13. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

6. Homework, Page 170 Find the derivative of each function. 17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

7. Homework, Page 170 Find the derivative of each function. 21. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

8. Homework, Page 170 Find the derivative of each function. 25. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

9. Homework, Page 170 Calculate the second derivative. 29. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

10. Homework, Page 170 Find an equation of the tangent line at the point specified. 33. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

11. Homework, Page 170 Find an equation of the tangent line at the point specified. 37. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

12. Homework, Page 170 Find an equation of the tangent line at the point specified. 37. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

13. Homework, Page 170 Verify the formula. 41. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

14. Homework, Page 170 43. Calculate the first five derivatives of f (x) = cos x. Then determine f (8) and f (37). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

15. Homework, Page 170 45. Calculate f ′(x) and f ′″(x) where f (x) = tan x Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

16. Homework, Page 170 49. The height at time t (s) of a weight, oscillating up and down at the end of a spring, is s(t) = 300 + 40 sin t cm. Find the velocity and acceleration at t = π/3 s. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

17. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 3: Differentiation Section 3.7: The Chain Rule

18. Composite Functions Composite functions are combinations of simpler functions, for instance f (x) = sin e x. None of the rules of differentiation we learned thus far permit us to differentiate a composite function. Remembering that the composition of f (x) and g (x) is written as f ○ g (x) or f (g (x)), we use the Chain Rule to differentiate composites. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

19. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If we choose, we may also represent a composite function of f (x) and u (x) as f (u). The chain rule then becomes: which may also be written as:

20. Example, Page 178 8. Calculate d/dx f (x 2 + 1) for the following choices of f (u ): Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

21. Example, Page 178 Find the derivative of f ○ g. 16. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company