1. PRODUCT & QUOTIENT RULES & HIGHER-ORDER DERIVATIVES (2.3)
September 25th, 2017

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. Proof of Thm. 2.7: The Product Rule
Prove

4. 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

5. Proof of Thm. 2.8: The Quotient Rule
Prove

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

7. III. DERIVATIVES OF TRIGONOMETRIC FUNCTIONS
Thm. 2.9: Derivatives of Trigonometric Functions:

8. A. PROOF OF THE DERIVATIVE OF SEC X
Prove

9. Ex. 1: Find each derivative.b) c)

10. Ex. 2: Use the graphs of f(x) and g(x) below to evaluate each of the derivatives.
Let

11. 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

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

13. Ex. 3: Given the position function , where t is measured in seconds and the position is given in feet, answer each of the following.
What is the instantaneous acceleration of the object after 10 seconds?
b) What is the average acceleration of the object from 2 to 10 seconds?
c) Are there any times when the instantaneous velocity is the same as the average velocity on the interval from 0 to 25 seconds?