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 ,