Chapter 15 Rules of Differentiation

1. Chapter 15Rules of Differentiation A Simple rules of differentiation B The chain rule C The product rule D The quotient rule E Derivatives of exponential functions F Derivatives of logarithmic functions G Derivatives of trigonometric functions H Second and higher derivatives

2. Opening Problem

3. Differentiation Differentiation is the process of finding a derivative or gradient function. Given a function f(x), we obtain f’(x) by differentiating with respect to the variable x.

4. General rules for differentiation

5. Example

11. Chain Rule

14. Product Rule

18. Quotient Rule

22. Derivatives of exponential functions If f(x) = bxthen f’(x) = kbxwhere k is a constant equal to f’(0).

23. If f(x) = ex then f’(x) = ex. For large negative x, f(x) = exapproaches the asymptote y = 0. ex > 0 for all x, so the range of f(x) = exis R+.

24. Find the gradient function for y equal to a. 2ex + e-3x b. x2e-x c.

31. Find the derivative for

32. Derivatives of logarithmic functions

37. Derivatives of trigonometric functions Think back to the light on a ferris wheel problem in chapter 10. The light was moving at a certain rate with respect to time. The velocity of the light is called the angular velocity because it was changing angle with respect to time. The angular velocity of the light (P) is the rate of change of angle AOP where A is the starting point of the light.

38. If we let l be the arc length AP, the linear speed of P is dl /dt, the rate of change of l with respect of time.

45. Second and higher derivatives

48. Nth derivative