1. Product & quotient rules & higher-order derivatives (2.3) October 17th, 2012

2. I. the product rule Thm. 2.7: The Product Rule: The product of two differentiable functions f and g is differentiable. The derivative of fg is the first function times the derivative of the second, plus the second function times the derivative of the first.

3. Ex.1: Find the derivative of each function.

4. You Try: Find the derivative of each function.

5. II. the quotient rule Thm. 2.8: The Quotient Rule: The quotient f/g of two differentiable functions f and g is differentiable for all value of x for which.

6. A. using the quotient rule Ex. 2: Find the derivative of.

7. You Try: Find the derivative of.

8. B. Rewriting before differentiating Ex. 3:Find the slope of the tangent line to the graph of at (-1, -7/3).

9. You Try: Find an equation of the tangent line to the graph of at (1, 3).

10. *If it is unnecessary to differentiate a function by the quotient rule, it is better to use the constant multiple rule.

11. c. using the constant multiple rule Ex. 4: Find the derivative of each function. (a) (b)

12. You Try: Find the derivative of each function. (a) (b)

13. III. derivatives of trigonometric functions Thm. 2.9: Derivatives of Trigonometric Functions:

14. A. Proof of the derivative of sec x Ex. 5: Prove.

15. B. differentiating trigonometric functions Ex. 6: Find the derivative of each function. (a) (b)

16. C. Different forms of a derivative Ex. 7:Differentiate both forms of.

17. IV. Higher-order derivatives *We know that we differentiate the position function of an object to obtain the velocity function. We also differentiate the velocity function to obtain the acceleration function. Or, you could differentiate the position function twice to obtain the acceleration function. s(t) position function v(t) = s’(t) velocity function a(t) = v’(t) = s’’(t) acceleration function

18. *Higher-order derivatives are denoted as follows: First derivative Second derivative Third derivative Fourth derivative. nth derivative

19. A. finding acceleration Ex. 8: Given the position function, find the acceleration at 5 seconds. (Let s(t) be in feet).

20. You Try: Given the position function, where s(t) is in feet, find that acceleration at 10 seconds.