Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Do Now: Aim: To memorize more stuff about differentiation: Product/quotient rules and.

Aim: Product/Quotient & Higher Order Derivatives

Course: Calculus

Do Now:

Aim: To memorize more stuff about differentiation: Product/quotient rules and more!!!!

Find the slope of the tangent at 3

8for ( )

9

x

g xx


Course: Calculus

Two Helpful Basics

2If , then

If , then 2

k dy ky

x dx xdy k

y k xdx x

3 3

2 28

Find '( ) ( ) 6 12 24f x f x x x xx

1

23

62

x

3

21

82

x

12

2 x

5

23

242

x

3 5

2 26

'( ) 9 4 36f x x x xx


Course: Calculus

The Product Rule

The derivative of the product of two differentiable functions f and g is itself differentiable. The derivative of fg is the first function times the derivative of the second plus the second function times the derivative of the first. d

f f x g x g x f xx g xdx

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

Find the derivative of h(x) = (3x – 2x2)(5 + 4x)

h x x x x x2'( ) (3 2 )(4) (5 4 )(3 4 )

h x x x x x2 2'( ) (12 8 ) (15 8 16 )

= -24x2 + 4x + 15

'( )h x 2(3 2 )x x

first derivative of second

[5 4 ]d

xdx

second

(5 4 )x

derivative of first

2[3 2 ]d

x xdx

( ) 'dv du

f x uv f x u vdx dx


Course: Calculus

The Quotient Rule

The derivative of the quotient of two differentiable functions f and g is itself differentiable at all values of x for which g(x) 0. The derivative of f/g is given by the denominator times the derivative of the numerator minus the numerator time the derivative of the denominator, all divided by the square of the denominator.

f xd g x f x f x g xg xdx g x

2

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

2( ) '

du dvv uu dx dxf x f x

v v


Course: Calculus

The Quotient Rule

f xd g x f x f x g xg xdx g x

2

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

Find the derivative of

2( 1)'

xy

denom.

2

5 21

xy

x

2

2 2

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

( 1)x x x

yx

2 2

2 2

(5 5) (10 4 )'

( 1)x x x

yx

2

2 2

5 4 5( 1)x xx

square of denom.2 2( 1)x

derivative of numer.

[5 2]d

xdx

numer.

(5 2)x

derivative of denom.

2[ 1]d

xdx


Course: Calculus

Model Problems

Find the derivative of3 (1/ )

5x

yx

3 (1/ )5x

yx

2

3 15

xy

x x

2

2 2

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

dy x x x xdx x x

2

2 2

3 2 5( 5 )x xx x

simplify

rewrite to eliminate complex nature


Course: Calculus

Model Problems

Rewriting Quotients to utilize the Constant Multiply Rule reduces work required.

2 36

x xy

21

36

y x x

rewrite to eliminate complex nature

1' 2 3

6y x differentiate

2 3'

6x

y

simplify


Course: Calculus

Model Problems

4 7 10

2

Find if 5 3 8

2 4Find if

6

dyy x x x x

dx

dy xy

dx x


Course: Calculus

Do Now:

Aim: To memorize more stuff about differentiation: Product/quotient rules and more!!!!

Find the derivative

3 8

2 6

x x

x x


Course: Calculus

Derivatives of Trig Functions

tand

xdx

cotd

xdx

Differentiate both sides individually1 cos

csc cotsin

xy x x

x

2

2 2

2 2

(sin )(sin ) (1 cos )(cos )1 cossinsin

sin cos cos 1 cossin sin

d x x x xxdx xx

x x x xx x

sec tanx x csc cotx x

2sec x 2csc x

cscd

xdx

secd

xdx

sintan

cosx

xx

left side


Course: Calculus

Derivatives of Trig Functions

Differentiate both sides individually1 cos

csc cotsin

xy x x

x

csc cotd

x xdx

Show two derivatives are equal

2csc csc cotx x x

right side2csc cot cscx x x

2

1 cos?

sinx

x

2 2

1 cos 1 1 cossin sin sin sin

x xx x x x


Course: Calculus

Higher Order Derivatives

First Derivative: y’, f’(x), , ( ) , x

dy df x D y

dx dx

Second Derivative: y’’, f’’(x),

2 22

2 2, ( ) , x

d y df x D y

dx dxThird Derivative: y’’’, f’’’(x),

3 33

3 3, ( ) , x

d y df x D y

dx dx

nth Derivative: y(n), f(n)(x), , ( ) ,

n nn

xn n

d y df x D y

dx dx

s’(t) = v(t) Velocity Function

s’’(t) = v’(t) = a(t) Acceleration Function

Position Function s t gt v t s20 0

1( )

2


Course: Calculus

Model Problem

Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by

s(t) = -0.81t2 + 2

where s(t) is the height in meters and t is the time in seconds. What is the ratio of the earth’s gravitational force to the moon’s?


Course: Calculus

Model Problem

s(t) = -0.81t2 + 2 Position Function

s’(t) = v(t) = -1.62t Velocity Function

s’’(t) = v’(t) = a(t) = -1.62 Acceleration Function

Acceleration due to gravity on the moon is -1.62 meters per second per second.

Acceleration due to gravity on earth is -9.8 meters per second per second.

Earth's gravitational force 9.86.05

Moon's graviational force 1.62


Course: Calculus

nDeriv(

A ball is thrown straight up into the air, and the height of the ball above the ground is given by the function h(t) = 6 + 37t – 16t2, where h is in feet and t is in seconds. What is the velocity of the ball at time t = 3.2?

MATH 8 x,ENTER 6 + 37t – 16t2 , 3.2

ENTER


Course: Calculus

Model Problem

22 1x Find the derivative

Find the derivative when x = 2 using nDeriv( function of calculator


Course: Calculus

Model Problem

2 1

3

x x

x

Find the derivative



Course: Calculus

Model Problem

33 4 5 1x x x x Find the derivative



Course: Calculus

Model Problem

3 3 1 2 at (1, 3)x x x

Find an equation of the tangent line to the graph

Use nDeriv( function of calculator


Course: Calculus

Model Problem

Find the derivative

5 cscy x x

2 sin 2 cosy x x x x

1 sin

1 sin

xy

x

Aim: Product/Quotient & Higher Order Derivatives Course: Calculus Do Now: Aim: To memorize more stuff about differentiation: Product/quotient rules and.

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