4.2:Derivatives of Products and Quotients

Objectives: Students will be able to…•Use and apply the product and quotient rule for differentiation

The Product Rule

The derivative of the product of 2 functions is the first function times the derivative of the second, plus the second function times the derivative of the first.

Let f(x) = u(x) v(x)∙ (u’(x) and v’(x) exist) f’(x) = u(x) v’(x) + v(x) u’(x)∙ ∙ Example: Find f’(x) if f(x) = (2x+3)(x2-4)

HOW ELSE COULD YOU HAVE DONE THIS?

USING THE PRODUCT RULE, FIND THE DERIVATIVE OF THE FOLLOWING FUNCTIONS.

1.

2.

3.

4.

27311

)( 32

x

xxxf

4323 287 xxxxy

xxxxp 21)( 22

423 xxy

QUOTIENT RULE

The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Let f(x) = , v(x) ≠ 0, v’(x) and u’(x) exist

f’(x) =

Be careful …Use parenthesis when subtracting function in numerator. Be aware of signs!

)(

)(

xv

xu

2)()(')()(')(

xv

xvxuxuxv

Find f’(x) if f(x) =24

13

x

x

Use the quotient rule to find the derivatives.

1.

2.

1

252

x

xy

1)(

2 x

xxf

Find the derivative.

1.

2.

1

1)(

2

2

t

tttf

2

3)(

xxh

Find the equation for the tangent line to the curve at (1,2)

x

xy

2

32

Applications of Derivative

• Position function: s(t)• Velocity: rate of change of the position with

respect to time: v(t) = s’(t)Velocity gives speed as well as direction• Speed : | v(t) |• Acceleration: rate of change of velocity with

respect to time: a(t) = v’(t) = s’’(t)

• An object is slowing down when…. velocity and acceleration are opposite signs

• An object is speeding up when….. velocity and acceleration are the same sign