AP CALCULUS

1008 : Product and Quotient Rules

PRODUCT RULE FOR DERIVATIVES

 Product Rule:

 

(In Words) ________________________________________________

( ) ( )y f x g x

2( ) (3 2 )(5 4 )f x x x x

y u v

The first times the derivative of the second plus the second times the derivative of the first.

→𝑢𝑣′+𝑣𝑢′𝑦 ′= 𝑓 (𝑥 )𝑔′ (𝑥 )+𝑔 (𝑥 ) 𝑓 ′ (𝑥)

f(x) g(x)

First*d(second)+second*d(first)

fi d(sec)+sec d(fi)

PRODUCT RULE FOR DERIVATIVES

2( ) (3 2 )(5 4 )f x x x x EX:

𝑦 ′=(3 𝑥− 2𝑥2 ) (− 4 )+(5 − 4 𝑥)(3 − 4 𝑥)

𝑦 ′=−12+8𝑥2+15 −20 − 12𝑥+16 𝑥2

𝑦 ′=24 𝑥2− 44 𝑥+15

Example Using the Product Rule

3 2Find if 4 3f x f x x x

3 2

3 2 3 2 2

4 4 2

4 2

Using the Product Rule with 4 and 3,gives

4 3 4 2 3 3

2 8 3 9

5 9 8

u x v xdf x x x x x x xdx

x x x x

x x x

GRADING:

PRODUCT RULE FOR DERIVATIVES

d uvdx

EX: Given u and v are differentiable at x = 2 and

u(2) = 3 u’(2) = 4 v(2) = - 2 v’(2) = 5

𝑢𝑣 ′+𝑣𝑢 ′=3 (5 )+−2 (4 )=7

QUOTIENT RULES FOR DERIVATIVES

If you can reduce the fraction - - - - - DO

1( ) xf xx

2 4 3( ) x xf xx

→ 𝑦=(𝑥+1 ) 𝑥− 1=𝑥𝑥 +

1𝑥=1+𝑥− 1

)

𝑦=𝑥32 +4 𝑥

12 +3 𝑥

− 12

QUOTIENT RULES FOR DERIVATIVES

Quotient Rule:

 

(In Words) ___________________________

¡ ¡ ¡ ¡ WATCH YOUR ALGEBRA ! ! ! !

22 4 3( )2 3

x xf xx

( )( )

f xyg x

uyv

𝑦 ′=𝑔 (𝑥 ) 𝑓 ′ (𝑥 )− 𝑓 (𝑥 )𝑔 ′(𝑥 )  

(𝑔 (𝑥 ))2

𝑦 ′=𝑣 𝑢′ −𝑢𝑣 ′𝑣2

𝑏𝑜𝑡𝑡𝑜𝑚∗𝑑 (𝑡𝑜𝑝 )− 𝑡𝑜𝑝∗𝑑 (𝑏𝑜𝑡𝑡𝑜𝑚)𝑏𝑜𝑡𝑡𝑜𝑚2

QUOTIENT RULES FOR DERIVATIVES

22 4 3( )2 3

x xf xx

EX:

𝑦 ′=(2− 3𝑥 ) (4 𝑥− 4 ) −(2𝑥2 −4 𝑥+3)(−3)

(2− 3𝑥 )2

𝑦 ′=8𝑥−8 − 12𝑥2+12𝑥+6 𝑥2−12 𝑥+9

(2− 3𝑥 )2

𝑦 ′=− 6 𝑥2+8 𝑥+1

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Example Using the Quotient Rule

3

2

4Find if 3

xf x f xx

3 2

3 2 2 3

22 2

4 2 4

22

4 2

22

Using the Quotient Rule with 4 and 3,gives

4 3 3 4 2

3 3

3 9 2 8 3

9 8

3

u x v x

x x x x xdf xdx x x

x x x x

x

x x x

x

GRADING:

QUOTIENT RULES FOR DERIVATIVES

2

1( )4 3

f xx x

EX:

QUOTIENT RULE FOR DERIVATIVES

d udx v

EX: Given u and v are differentiable at x = 2 and

u(2) = 3 u’(2) = 4 v(2) = - 2 v’(2) = 5

Last Update

• 9/17/11

Assignment: p 124 # 1 - 41 odd