Mastering Product and Quotient Rules in Calculus

1. Calculus, Section 3.3

2. Product and Quotient Rules This section covers the Product Rule and the Quotient Rulefor computing derivatives. These two rules, together with the Chain Rule and implicit differentiation (covered in later sections), make up an extremely effective “differentiation toolkit.”

3. Product Rule We will consider the Product Rule first. It may be helpful to remember the Product Rule in words: The derivative of a product is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function:

4. Product Rule (Instructor’s Note) Personally, I remember the Product Rule using the following mnemonic Keep the first, differentiate the second. (plus) Keep the second, differentiate the first.

5. Product Rule In Leibniz notation, the Product Rule can be written as follows: Another notation is as follows:

6. Instructor’s Note When calculating derivatives using the Product Rule (and the Quotient Rule later), it helps to pull off to the side and calculate the derivative of each function separately so you can “plug in” the resulting expression.

7. Example 1 Find the derivative of .

8. Example 2 Find the derivative of .

9. Example 3 Calculate .

10. Quotient Rule The next theorem states the rule for differentiating quotients. Note, in particular that is not equal to the quotient . You can think of the quotient rule using the following memory aid:

11. Quotient Rule In Leibniz notation, the Quotient Rule can be written as follows: Another notation is as follows:

12. Example 4 Compute the derivative of .

13. Example 5 Calculate .