2.3 PRODUCT & QUOTIENT RULES AND HIGHER-ORDER DERIVATIVES
Ms. Clark10/4/2016
WARM UP [FROM 1998 AP CALCULUS EXAM NO CALCULATOR]
An equation of the line tangent to the graph of 𝑦𝑦 = 𝑥𝑥 + cos 𝑥𝑥 at the point (0,1) is :
(A) 𝑦𝑦 = 2𝑥𝑥 + 1 (B) 𝑦𝑦 = 𝑥𝑥 + 1 (C) 𝑦𝑦 = 𝑥𝑥 (D) 𝑦𝑦 = 0
MORE SHORTCUTS!Based on the rules we have learned so far, how would you find the equation of the tangent line at the point 𝑥𝑥 = 0 for the following function?
1a.) ℎ 𝑥𝑥 = (𝑥𝑥2 + 𝑥𝑥 − 1)(2𝑥𝑥3 + 6𝑥𝑥2 − 7𝑥𝑥 + 4)
LET’S SEE IF WE CAN FIND A SHORTCUT 1b.) ℎ 𝑥𝑥 = (𝑥𝑥2 + 𝑥𝑥 − 1)(2𝑥𝑥3 + 6𝑥𝑥2 − 7𝑥𝑥 + 4)ℎ 𝑥𝑥 is a product of two functions. We’ll call them 𝑓𝑓 𝑥𝑥 and 𝑔𝑔 𝑥𝑥
ℎ 𝑥𝑥 = 𝑓𝑓(𝑥𝑥) � 𝑔𝑔(𝑥𝑥)
Now we can do this problem: Based on the rules we learned so far, how would you find the equation of the tangent line at the point where 𝑥𝑥 = 0 for the following function?
1c) ℎ 𝑥𝑥 = (𝑥𝑥2 + 𝑥𝑥 − 1)(2𝑥𝑥3 + 6𝑥𝑥2 − 7𝑥𝑥 + 4)
PRODUCT RULE
If 𝑓𝑓 𝑥𝑥 = 𝑔𝑔 𝑥𝑥 � ℎ 𝑥𝑥 then 𝑓𝑓′ 𝑥𝑥 =
Using the product rule find the derivative:
2. ) 𝑦𝑦 = 𝑥𝑥(𝑥𝑥 − 1)
Using the product rule find the derivative:
3.) ℎ 𝑥𝑥 = 2𝑥𝑥3 sin 𝑥𝑥 ℎ′ 𝑥𝑥 =
4.) 𝑓𝑓 𝑥𝑥 = 2 cos 𝑥𝑥 sin 𝑥𝑥 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
=
5a.) Use the definition of a derivative to find the derivative of 1𝑔𝑔 𝑑𝑑
5b.) 𝑓𝑓 𝑑𝑑𝑔𝑔 𝑑𝑑
′?
QUOTIENT RULE
If 𝑓𝑓 𝑥𝑥 = 𝑔𝑔 𝑑𝑑ℎ 𝑑𝑑
then 𝑓𝑓′ 𝑥𝑥 =
Using the quotient rule, find the derivative
6) 𝑓𝑓 𝑥𝑥 = 𝑑𝑑+14𝑑𝑑+5
𝑓𝑓′ 𝑥𝑥 =
7) 𝑓𝑓 𝑥𝑥 = 𝑑𝑑2+𝑑𝑑−1𝑑𝑑3
𝑓𝑓′ 𝑥𝑥 =
8) 𝑓𝑓 𝑥𝑥 = 18𝑑𝑑2
cos 𝑑𝑑𝑓𝑓′ 𝑥𝑥 =
9) 𝑓𝑓 𝑥𝑥 = 18𝑑𝑑2
cos 𝑑𝑑 sin 𝑑𝑑𝑓𝑓′ 𝑥𝑥 =
More TRIG derivatives
10.) 𝑓𝑓 𝑥𝑥 = tan 𝑥𝑥 𝑓𝑓′ 𝑥𝑥 =
11.) 𝑓𝑓 𝑥𝑥 = cot 𝑥𝑥 𝑓𝑓′ 𝑥𝑥 =
More TRIG derivatives
12.) g 𝑥𝑥 = sec 𝑥𝑥 𝑔𝑔′ 𝑥𝑥 =
13.) q 𝑥𝑥 = c𝑠𝑠𝑠𝑠 𝑥𝑥 𝑞𝑞′ 𝑥𝑥 =
HIGHER ORDER DERIVATIVES
14.) 𝑓𝑓 𝑥𝑥 = 𝑥𝑥5
Find 𝑓𝑓′ 𝑥𝑥 , 𝑓𝑓′′ 𝑥𝑥 , 𝑓𝑓4 𝑥𝑥 , 𝑓𝑓5 𝑥𝑥 , 𝑓𝑓6(𝑥𝑥)
HIGHER ORDER DERIVATIVES
15.) 𝑓𝑓 𝑥𝑥 = 2𝑥𝑥3 − 2𝑑𝑑
Find 𝑓𝑓′ 𝑥𝑥 , 𝑓𝑓′′ 𝑥𝑥 , 𝑓𝑓4 𝑥𝑥 , 𝑓𝑓5 𝑥𝑥 ,