1. 4. Product and Quotient Rules

2. Product Rule
Multiplication and addition are both commutative so the order does not matter

3. Example 1
Find the derivative using the product rule, then verify using the power rule and graphically using NDERIV.

4. Example 2
Find the derivative of each of the following

5. Quotient Rule
Subtraction is not commutative so this one is more important to get straight
If you call the numerator “HI” and the denominator “LO”, and d is derivative, this becomes

6. Example 3
Find the derivative using the quotient rule, then verify graphically using NDERIV.

7. Example 4
Rewriting is still key. Find the derivative of each of the following using the most efficient method

8. Example 5
Find the derivative of using the quotient rule.

9. Derivatives of other trig functions

10. Example 6
Find the derivatives of each of the following

11. Example 7
Find the derivatives of each of the following, then show they are equivalent.

12. Higher order derivatives
If you take the derivative twice it is called the second derivative. You can continue to take derivatives as long as you like.

13. Example 8
Find the first 5 derivatives of

14. What information does the derivative at a point tell us?
Tells us whether the tangent line has a positive or negative slope
Tells us how steep the line is (the larger the derivative, the steeper the line)
Tells us if there is a turning point (slope is 0)

15. Example 9
The graph below is the graph of f. On the same coordinate grid, sketch and label the possible graphs of f’, f’’, f’’’.