1. Derivatives of polynomials Derivative of a constant function We have proved the power rule We can prove

2. Rules for derivative The constant multiple rule: The sum/difference rule:

3. Exponential functions Derivative of The rate of change of any exponential function is proportional to the function itself. e is the number such that Derivative of the natural exponential function

4. Product rule for derivative The product rule: g is differentiable, thus continuous, therefore,

5. Remark on product rule In words, the product rule says that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function. Derivative of a product of three functions:

6. Example Find if Sol.

7. Quotient rule for derivative The quotient rule:

8. Example Using the quotient rule, we have: which means is also true for any negative integer k.

9. Homework 4 Section 2.7: 8, 10 Section 2.8: 16, 17, 22, 24, 36 Section 2.9: 28, 30, 46, 47 Page 181: 13

10. Example We can compute the derivative of any rational functions. Ex. Differentiate Sol.

11. Table of differentiation formulas

12. An important limit Prove that Sol. It is clear that when thus Since and are even functions, we have Now the squeeze theorem together with gives the desired result.

13. Derivative of sine function Find the derivative of Sol. By definition,

14. Derivative of cosine function Ex. Find the derivative of Sol. By definition,

15. Derivatives of trigonometric functions Using the quotient rule, we have:

16. Change of variable  The technique we use in is useful in finding a limit.  The general rule for change of variable is:

17. Example Ex. Evaluate the limit Sol. Using the formula and putting u=(x-a)/2, we derive

18. Example Ex. Find the limit Sol. Using the trigonometry identity and putting u=x/2, we obtain

19. Example Ex. Find the limits: (a) (b) Sol. (a) Letting then and (b) Letting then