Lesson #25
Product Rule

Quote of the Day:
"We make a living by what we get, but we make a life by what we give." -- Winston Churchill

Objectives:
The student will prove the product rule for derivatives.

The student will apply the product rule to differentiation problems.

1. Bellringer.

2. To introduce the Product Rule:
First, list rules for differentiation that we have already proven:

(A) Definition of the derivative
(B) Derivative of a constant
(C) Power Rule
(D) Derivative of a sum or difference

Recall some of the rules the student has already learned:

(A) The limit of a sum is the sum of the limits.

(B) The limit of a product is the product of the limits.

(C) The derivative of a sum is the sum of the derivatives.

(D) The derivative of a difference is the difference of the derivatives.

So, many calculus students conjecture that the derivative of a product is the product of the derivatives.   Have the students test this conjecture with a few examples:

3. Example #1


4. Testing the conjecture on Example #2:



5. Proof of the PRODUCT RULE


6. Extended Product Rule:
(f g h)' = f' g h + f g' h + f g h'
(f g h j)' =

7. Example:
Given y = (5x² - 3) (7x³ + x), Find y'
Answer: y' = 175x⁴ - 48x² - 3

8. Assignment: p. 203 (1, 3, 5, 9, 19a, 21a,b, 33)

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Send any comments or questions to: David Pleacher