Aim: Implicit Differentiation Course: Calculus Do Now: Aim: What Is Implicit Differentiation and How Does It Work? Explicit Implicit

Aim: Implicit Differentiation Course: Calculus

Do Now:

Aim: What Is Implicit Differentiation and How Does It Work?

2 3Find the derivative of 4 3 1x y y x

2 34 3 1y x x

3

2

1

4 3

xy

x

4 2

22

4 9 8'

4 3

x x xy

x

Explicit

Implicit


Implicit vs. Explicit

Explicit Form 1

yx

derivative of y?

variable y is written as a function of x1 2'y x y x

Often you can solve for y in term of x

Not Always!2 32 4 2x y y

Implicit Differentiation is used

1xy

Implicit Form


Differentiating with Respect to x

3 2. 3xx

da x

d

variables agree

Use Simple Power Rule

3. ydx

db

variables disagree

Use Chain Rule

un u’nun-1

. 3d

c x ydx

Chain Rule 3 3 'd

y ydx

2.d

d xydx

Product Rule

Chain Rule

Simplify

23 ydy

dx

1 3dy

dx

2 2d dx y y x

dx dx

2 22 1 2dy dy

x y y xy ydx dx


Differentiating with Respect to x

3 2. 3xx

da x

d

variables agree

Use Simple Power Rule

3. ydx

db

variables disagree

Use Chain Rule

un u’nun-1

. 3d

c x ydx

Chain Rule 3 3 'd

y ydx

2.d

d xydx

Product Rule

Chain Rule

Simplify

23 ydy

dx

1 3dy

dx

2 2d dx y y x

dx dx

2 22 1 2dy dy

x y y xy ydx dx

COMMON ERROR! DON’T FORGET

6 56dy

yd dd

xy

x


Guidelines for Implicit Differentiation

1. Differentiate both sides of the equation with respect to x.

2. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation.

3. Factor dy/dx out of the left side of the equation.

4. Solve for dy/dx by dividing both sides of the equation by the left-hand factor that does not contain dy/dx.


Model Problem

Find dy/dx given y3 + y2 – 5y – x2 = -4

1. Differentiate both sides of the equation with respect to x.

3 2 2

3 2 2

2

5 4

5 4

3 2 5 2 0

d dy y y x

dx dxd d d d d

y y y xdx dx dx dx dx

dy dy dyy y x

dx dx dx

2. Collect all terms involving dy/dx on the left side of the equation

23 2 5 2dy dy dy

y y xdx dx dx


Model Problem

Find dy/dx given y3 + y2 – 5y – x2 = -43. Factor dy/dx out of the left side of the equation.

4. Solve for dy/dx by dividing by (3y2 + 2y – 5)

2

2

3 2 5

dy x

dx y y

23 2 5 2dy

y y xdx

function? NO

y3 + y2 – 5y – x2 = -4

(1, 1)(2, 0)

(1, -3)

slope at (1, 1)slope at (2, 0)slope at (1, -3)

und-4/51/8


Functions from Equations

If a segment of a graph can be represented by a differentiable function, dy/dx will have meaning as the slope.

1.5

1

0.5

-0.5

-1

-1.5

-1 1

function? NO2 2 1x y

YES

YES21y x

21y x

Recall: a function is not differentiable at points with vertical tangents nor at points where the function is not continuous


Do Now:

Aim: What Is Implicit Differentiation and How Does It Work?

Determine the slope of the tangent line to the graph x2 + 4y2 = 4 at the point

.1

2,2


Model Problem

Determine the slope of the tangent line to the graph x2 + 4y2 = 4 at the point

.1

2,2

implicit differentiation2 8 ' 0x yy

solve for dy/dx2

'8 4

x xy

y y

evaluate for the point

Slope of tangent at is 1/2 2, 1 / 2

Note: 'dy

ydx


Model Problem

Determine the slope of the tangent line to the graph 3(x2 + y2)2 = 100xy and the point (3, 1).

22 23 100d d

x y xydx dx

2 23 2 2 2 ' 100 ' 1x y x yy xy y

Constant and General Power Rules

2 26 2 ' 100 ' 100x y x yy xy y

2 212 ' 100 ' 100x y x yy xy y

FOIL and isolate dy/dx


Model Problem


2 2 2 212 ' ' 100 ' 100x x x yy y x y yy xy y

2 2 2 212 ' ' 100 ' 100x x y x x yy y yy xy y

2 2 2 212 ' 100 ' 100x x y yy x y xy y

2 2 2 212 12 ' 100 ' 100x x y yy x y xy y

2 2 2 212 ' 100 ' 100 12yy x y xy y x x y


Model Problem


2 2 2 212 ' 100 ' 100 12yy x y xy y x x y

2 2 2 2' 12 100 100 12y y x y x y x x y

2 2

2 2

100 12'

12 100

y x x yy

y x y x

substitute (3, 1)

2 2

2 2

100 12 13'

91

1 3 3 1

1 3 1 32 100y


Finding the 2nd Derivative Implicitly

Given x2 + y2 = 25, find 2

2

d y

dx

2

2 2

1dy

y xd y dxdx y

find first derivative implicitly:

2 2dy

y xdx

2

2

dy x x

dx y y

2 2 0

dyx y

dx

quotient rule

2

xy x

y

y

sub –x/y for dy/dx

2 2

3

y x

y

sub 25 for x2+y2

2

2 3

25dy

dx y


Model Problem

Find the tangent line to the graph given by x2(x2 + y2) = y2 at the point 2 2

,2 2

4 2 2 2 0x x y y

3 2 24 2 ' 2 2 ' 0x x y y xy yy

3 2 24 2 2 ' 2 ' 0x xy yx y yy

2 2 22 2 ' 2 1 0x x y y y x

2 2

2

2 2'

2 1

x x yy

y x

implicit differentiation


Model Problem

Find the tangent line to the graph given by x2(x2 + y2) = y2 at the point 2 2

,2 2

2 2

2

2 2'

2 1

x x yy

y x

2 2

2

2

1

x x y

y x

2 2

2

2 2 22 2 2

2 22

'

12

2

y

= 3 = m

2 23 3 2

2 2y x y x

substitute

point-slope formula for equation (y – y1) = m(x – x1)


Model Problem

Find dy/dx implicitly for the equation sin y = x. Then find the largest interval of the form –a < y < a such that y is a differentiable function of x. sin

d dy x

dx dx

cos 1dy

ydx

1

cos

dy

dx y

6

4

2

-2

-4

-6

-8

-2 2

r y = sin y

1,2

1,2

explicitly:

for the interval -/2 < y < /2, we use and substitute the original equation to arrive at

2cos 1 siny y

2

1

1

dy

dx x

Documents

Aim: Implicit Differentiation Course: Calculus Do Now: Aim: What Is Implicit Differentiation and How Does It Work? Explicit Implicit