Product and Quotient Rules

Product Rule

• Many people are tempted to say that the derivative of the product is equal to the product of the derivatives (similar to addition and subtraction), but consider– What is product of derivatives?

– What is derivative of products?

– Are they equal?

2 and xxgxxf

Product Rule

• Theorem 3.4.1 If f and g are differentiable at x, then so is the product of , and

• In words, the derivative of a product is:

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

gf

)]([)()]([)()]()([ xfdx

dxgxg

dx

dxfxgxf

dx

d

Example 1

• Find dy/dx if )7)(14( 32 xxxy

2) Find if dt

ds tts 1

Quotient Rule

• Theorem 3.4.2 If f and g are differentiable at x and

, then f/g is differentiable at x and

• In words the derivative of a quotient is

Low d-high minus high d-low all over the denominator squared.

0)( xg

2)]([

)]([)()]([)(

)(

)(

xg

xgdxd

xfxfdxd

xg

xg

xf

dx

d

Example 3

• Find for 5

12 23

x

xxyy

Example 4

• By differentiating, find the exact locations of the horizontal tangent lines. Then check by graphing.

1

12

x

xxf

Practice

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