1. 2.3 The Product Rule
Notice the order doesn’t matter!
If ℎ 𝑥 =𝑓 𝑥 𝑔(𝑥), then ℎ ′ 𝑥 = 𝑓 ′ 𝑥 𝑔 𝑥 +𝑓 𝑥 𝑔′(𝑥).
ℎ ′ 𝑥 = lim ℎ→0 𝑓 𝑥+ℎ 𝑔(𝑥+ℎ)−𝑓 𝑥 𝑔(𝑥) ℎ
Proof:
Example: Differentiate 𝑓 𝑥 =( 𝑥 2 −3𝑥)( 𝑥 5 +2) using the product rule.
= lim ℎ→0 𝑓 𝑥+ℎ 𝑔 𝑥+ℎ −𝑓 𝑥 𝑔 𝑥+ℎ +𝑓 𝑥 𝑔(𝑥+ℎ)−𝑓 𝑥 𝑔(𝑥) ℎ
= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 𝑔 𝑥+ℎ +𝑓 𝑥 [𝑔(𝑥+ℎ)−𝑔(𝑥]) ℎ
= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ 𝑔 𝑥+ℎ +𝑓 𝑥 𝑔(𝑥+ℎ)−𝑔(𝑥) ℎ
= lim ℎ→0 𝑓 𝑥+ℎ −𝑓 𝑥 ℎ lim ℎ→0 𝑔 𝑥+ℎ + lim ℎ→0 𝑓 𝑥 lim ℎ→0 𝑔(𝑥+ℎ)−𝑔(𝑥) ℎ
= 𝑓 ′ 𝑥 𝑔 𝑥 +𝑓 𝑥 𝑔′(𝑥).

2. Practice
Find the value ℎ ′ −1 for the function ℎ 𝑥 =(5 𝑥 3 +7 𝑥 2 +3)(2 𝑥 2 +𝑥+6) −𝟖

3. In summary …
We learned another rule to help us determine the derivative more efficiently - THE PRODUCT RULE!
QUESTIONS: p #1, 2, 3cdf, 6, 7a, 10 (Find the equation of the normal)