Upload
victor-beasley
View
216
Download
3
Embed Size (px)
Citation preview
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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?
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
Course: Calculus
Model Problem
22 1x Find the derivative
Find the derivative when x = 2 using nDeriv( function of calculator
Aim: Product/Quotient & Higher Order Derivatives
Course: Calculus
Model Problem
2 1
3
x x
x
Find the derivative
Find the derivative when x = 3 using nDeriv( function of calculator
Aim: Product/Quotient & Higher Order Derivatives
Course: Calculus
Model Problem
33 4 5 1x x x x Find the derivative
Find the derivative when x = 3 using nDeriv( function of calculator
Aim: Product/Quotient & Higher Order Derivatives
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
Aim: Product/Quotient & Higher Order Derivatives
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