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The Product and Quotient Rules and Higher Order Derivatives

Section 2.3

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After this lesson, you should be able to:

Find the derivative of a function using the Product RuleFind the derivative of a function using the Quotient RuleFind the derivative of a trig functionFind a higher-order derivative of a function

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The Product Rule

d dv duu v u v

dx dx dx

Example: Find dy/dx :

2( 6 )(3 2)y x x x

( )u f x ( )v g xLet f and g be differentiable functions and let and

' 'uv vu

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Product Rule Example

Example: Find dy/dx :

xxxxy cossin23

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Product Rule Example

Example: Find f ’(x):

)1)(32()( xxxf

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Product Rule Example

Example: Find y’ :

)9)(43( 23 xxxy

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The Quotient Rule

2

v du dv

ud u dx dxdx v v

2

''

v

uvvu

Let f and g be differentiable functions such that and let and

( ) 0,g x ( )v g x( )u f x

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Quotient Rule Example

1) Find y ’: 2

3 1

2 5

xy

x

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Quotient Rule Example

2) Find f ’(x):

12

( )3xf x

x

10

3) Find2 3( 4)( 4 )

given2 3

dy x x xy

dx x

Quotient Rule Example

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4) Find3

3given

1

dy

d

x

xy

x

Quotient Rule Example

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5) Find3

2

'( ) given ( )5

xf x

xf x

Quotient Rule Example

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sin cosd

x xdx

cos sind

x xdx

tand

xdx

The Derivative of Tangent

The derivatives of the four remaining trig functions can be found using the derivatives of sine and/or cosine and the Quotient Rule.

For example,

14

1csc

d dx

dx dx

The Derivative of Cosecant

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The Derivatives of the Six Basic Trig Functions

2

2

sin cos

cos sin

tan sec

csc csc cot

sec sec tan

cot csc

dx x

dxd

x xdxd

x xdxd

x x xdxd

x x xdxd

x xdx

Now for fun, you can use the Quotient Rule to find the derivatives of the two remaining trig functions!

Here's a summary of the derivatives of the six trigonometric functions:

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Trig Derivative Example

Find the derivative of ( ) 5 sin tanf x x x x x

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Example: ( ) tanf x x x

Write the equation of the tangent line at 4

x

Equation of the Tangent Line

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Higher Order Derivatives

y =f(x)

')(' ydx

dyxf Ex:

83)( 24 xxxf

xxxf 64)(' 3

'')(''2

2

ydx

ydxf Ex: 612)('' 2 xxf

''')('''3

3

ydx

ydxf xxf 24)(''' Ex:

)4(4

4)4( )( y

dx

ydxf Ex: 24)()4( xf

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Higher Order Derivatives of sin x

xy sin

xdx

dycos

xdx

ydsin

2

2

xdx

ydcos

3

3

xdx

ydsin

4

4

xdx

ydcos

5

5

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Higher Order Derivatives Example

1) Find the second derivative of 5

( ) sin2

xf x x

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Higher Order Derivatives Example

2) Given 6 4 21 2( ) 5 3

30 3f x x x x x

a) Find ''( )f x

b) Determine the point(s) at which the function has a horizontal tangent.

''( )f x

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Homework

Section 2.3 page 126 #1-23 odd, 29, 39-43 odd, 51, 63, 67, 73, 99, 101, 116