Product and Quotient Rules and Higher Order Derivatives Section 2.3

“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.”

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

“Low d high minus high d low over the square of what’s below” Quotient Rule Find the derivative of 𝑦= 5𝑥−2 𝑥 2 +1 . 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”

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

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

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).