product & quotient rule

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HIGHER ORDER DERIVATIVES Product & Quotient Rule

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Product & Quotient Rule. Higher Order Derivatives. The Product Rule. Theorem. Let f and g be differentiable functions. Then the derivative of the product fg is (fg) '(x) = f(x) g '(x) + g(x) f '(x) - PowerPoint PPT Presentation

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HIGHER ORDER DERIVATIVES

Product & Quotient Rule

The Product Rule

Theorem. Let f and g be differentiable functions. Then the derivative of the product fg is

(fg) '(x) = f(x) g '(x) + g(x) f '(x)

In other words, first times the derivative of the second plus second times the derivative of the first.

Using the Product Rule

Example:2( ) (3 2 )(5 4 )h x x x x

2 2

2

2 2

2

(3 2 ) (5 4 ) (5 4 ) (3 2 )

(3 2 )(4) (5 4 )(3 4 )

(12 8 ) (15 20 12 16 )

24 4 15

d dx x x x x x

dx dx

x x x x

x x x x x

x x

Product Rule

Example:

3

3 3

3 2

2 3

( ) cos

cos cos

( sin ) cos (3 )

3 cos sin

f x x x

d dx x x xdx dx

x x x x

x x x x

Product Rule

Example

1

2

( ) sin

1(cos ) sin

2

sincos

22 cos sin

2

g x x x

x x x x

xx x

xx x x

x

Quotient Rule

Theorem. Let f and g be differentiable functions. Then the derivative of the quotient f/g is

2

( ) ( ) '( ) ( ) '( )

( ) ( )

d f x g x f x f x g x

dx g x g x

In other words, low d high minus high d low over low low.

Quotient Rule

ExampleFind the derivative of 2

5 2

1

xy

x

2

2 2

2 2

2 2

2 2

2 2

2

2 2

( 1)(5) (5 2)(2 )

( 1)

(5 5) (10 4 )

( 1)

5 5 10 4

( 1)

5 4 5

( 1)

x x x

x

x x x

x

x x x

x

x x

x

Rewriting Before Differentiating

Example:

2

2

2 2

2 2

2 2

2 2

2 2

2

2 2

13

( )5

13

( ) get rid of the fraction in the denom( 5)

3 1( )

5

( 5)(3) (3 1)(2 )'( )

( 5)

3 15 (6 2 )'( )

( 5)

3 15 6 2'( )

( 5)

3 2 15'( )

( 5)

xf xx

xx

f xx x

xf x

x

x x xf x

x

x x xf x

x

x x xf x

x

x xf x

x

Derivatives of Trigonometric Functions

2 2

sin cos cos sin

tan sec cot csc

sec sec tan csc csc cot

d dx x x x

dx dx

d dx x x x

dx dx

d dx x x x x x

dx dx

Proof of Derivative of Tangent

Considering sin

tancos

xx

x

2

2 2

2

2

2

(cos )(cos ) ((sin )( sin )tan

cos

cos sin

cos1

cos

sec

d x x x xx

dx x

x x

x

x

x

Differentiating Trig Functions

y = x – tan x

y = x sec x y’ = x(sec x tan x) + sec x (1)

= sec x (x tan x + 1)

2 2' 1 sec tany x x

Differentiating Trigonometric Functions

2

2 2

2

2

2 2

2

1 cos

sinsin (sin ) (1 cos )(cos )

'sin

sin cos cos

sin1 cos

sin1 cos

sin sin

csc csc cot

xy

xx x x x

yx

x x x

xx

xx

x x

x x x

Higher Order Derivatives

Applications:

Finding Acceleration Due to Gravity

Population Growth p. 125 problem 79

Any time we are asked to find the rate at which a rate is changing, this is a second derivative.