1. Product and Quotient Rules and Higher Order Derivatives
Section 2.3

2. “First d second plus second d first.”
Product Rule
Find the derivative of 𝑦=𝑥 cos 𝑥 .
Product Rule
𝑑 𝑑𝑥 𝑓 𝑥 𝑔(𝑥) =𝑓 𝑥 𝑔 ′ 𝑥 + 𝑔(𝑥)𝑓 ′ (𝑥)
“First d second plus second d first.”

3. Use the Product Rule
Use the product rule, when necessary, to calculate each derivative.
𝑦=(3𝑥−2 𝑥 2 )(5+4𝑥)
𝑦=2 𝑥 2 cos 𝑥 −4𝑥 sin 𝑥

4. “Low d high minus high d low over the square of what’s below”
Quotient Rule
Find the derivative of 𝑦= 5𝑥−2 𝑥
Quotient Rule
𝑑 𝑑𝑥 𝑓(𝑥) 𝑔(𝑥) = 𝑔 𝑥 𝑓 ′ 𝑥 −𝑓(𝑥) 𝑔 ′ (𝑥) 𝑔(𝑥) 2
“Low d high minus high d low over the square of what’s below”

5. Use the Quotient Rule Find each derivative: 𝑓 𝑥 = 𝑥 2 cos 𝑥
𝑔 𝑥 = sin 𝑥 𝑥 2 −5𝑥+2

6. Trig Function Derivatives
𝑑 𝑑𝑥 tan 𝑥 = 𝑠𝑒𝑐 2 𝑥 𝑑 𝑑𝑥 cot 𝑥 =− 𝑐𝑠𝑐 2 𝑥 𝑑 𝑑𝑥 sec 𝑥 = sec 𝑥 tan 𝑥 𝑑 𝑑𝑥 csc 𝑥 =− csc 𝑥 cot 𝑥

7. Higher Order Derivatives
Calculate each of the following:
𝑓 𝑥 =2 𝑥 2 −5𝑥−2, Find 𝑓 ′ (𝑥) and 𝑓 ′′ (𝑥).
𝑦=2 cos 𝑥 −5𝑥, Find 𝑑𝑦 𝑑𝑥 and 𝑑 2 𝑦 𝑑 𝑥 2 .
𝑦= sec 𝑥 , Find 𝑦 ′ and 𝑦 ′′ .
𝑓 𝑥 =6 𝑥 6 −2 𝑥 4 +4 𝑥 2 −7, Find 𝑓 5 (𝑥).