MAT 1221Survey of Calculus

Section B.1, B.2

Implicit Differentiation, Related Rates

http://myhome.spu.edu/lauw

Expectations

Use equal signs Simplify answers Double check the algebra

HW

WebAssign HW B.1, B.2 Additional HW listed at the end of the

handout (need to finish, but no need to turn in)

Need to do your homework soon. Do not wait until tomorrow afternoon.

Exam 1

Bring your Tutoring Bonus paper to class on Thursday.

Exam 1

You should have already started reviewing for Exam 1

Proficiency: You need to know how to do your HW problem on your own

You need to understand how to solve problems

Memorizing the solutions of all the problems is not a good idea

Preview

Extended Power Rule Revisit The needs for new differentiation

technique – Implicit Differentiation The needs to know the relation between

two rates – Related Rates

Extended Power Rule

dx

dunu

dx

dy

xguuy

xgy

n

n

n

1

)( ,

)(

dx

dunuu

dx

d nn 1

Extended Power Rule

dx

dunu

dx

dy

xguuy

xgy

n

n

n

1

)( ,

)(

dx

dunuu

dx

d nn 1

Extended Power Rule

dx

dunu

dx

dy

xguuy

xgy

n

n

n

1

)( ,

)(

dx

dunuu

dx

d nn 1

Extended Power Rule

dx

dunuu

dx

d nn 1

We now free the variable, which we need for the next formula.

Extended Power Rule

If is a function in , then

1n nd un

xu

xu

d

d d

If y is a function in x, then

dx

dynyy

dx

d nn 1

Extended Power Rule

If is a function in , then

If is a function in , then

1n nd yn

xy

xy

d

d d

1n nd un

xu

xu

d

d d

Example 0

dx

dynyy

dx

d nn 1

5dy

dx

The Needs for Implicit Differentiation…

Example 1

Find the slopes of the tangent line on the graph

i.e. find

122 yx

x

y

),( yxdx

dy

Example 1: Method I

Make as the subject of the equation:

122 yx

x

y

2

22

1

1

xy

xy

21 xy

Example 1: Method I

Make as the subject of the equation:

122 yx

x

y

2

22

1

1

xy

xy

21 xy

Example 1: Method I

Make y as the subject of the equation:

122 yx

x

y

2

22

1

1

xy

xy

21 xy

Example 1: Method I

Suppose the point is on the upper half circle

122 yx

x

y1

2 2 2

12 22

2

2

1 (1 )

1(1 ) (1 )

21

(0 2 )2 1

1

y x x

dy dx x

dx dx

xx

x

x

),( yx

Example 1: Method I

Suppose the point is on the lower half circle

122 yx

x

y1

2 2 2

12 22

2

2

1 (1 )

1(1 ) (1 )

21

(0 2 )2 1

1

y x x

dy dx x

dx dx

xx

x

x

),( yx

Example 1: Method I

Two disadvantages of Method I:

1. ???

2. ???

122 yx

Example 1: Method II

Implicit Differentiation:

Differentiate both sides of the equation.

122 yx

dx

dynyy

dx

d nn 1

2 2 1d dx y

dx dx

Expectation

You are required to show the implicit differentiation step

2 2 1d dx y

dx dx

Notations

(…) is the derivative of (…). Do not confuse it with which is the derivative of .

2 2 1x yd d

dx dx

Example 2

Find the slope of the tangent line at

xyyx 233

dx

dynyy

dx

d nn 1

1,1

1,1

3 3 2d dx y xy

dx dx

Notations

(1,1)

dym

dx

Notations

Correct or

Incorrect

because

2

2, 1,1

2 1 3 1

3 1 2 1x y

dy

dx

2

2

2 1 3 1

3 1 2 1

dy

dx

2

21,1

2 1 3 1

3 1 2 1

dy

dx

2

2

2 3

3 2

dy y x

dx y x

B.2. Related Rates

Related Rates

If and are related by an equation, their derivatives (rate of changes)

and

are also related.

Related Rates

If and are related by an equation, their derivatives (rate of changes)

and

are also related. Note that the functions are time dependent Extended Power Rule will be used

frequently, e.g.

2 5 42 ; 5t t t

d dx

t

d dyx x y y

d d d d

Example 3

Consider a “growing” circle.

Example 3

Both the radius and the area are increasing.

( )r t ( )A t

Example 3

What is the relation between and ?

( )r t ( )A t

Example 1

GO NUTS!

Example 1

GO N

UTS!

Example 3

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 1 Draw a diagram

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 2: Define the variables

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 3: Write down all the information in terms of the variables defined

A rock is thrown into a still pond and causes a circular ripple. If the radius of the ripple is increasing at 3 feet per second, how fast is the area changing when the radius is 5 feet?

Step 4: Set up a relation between the variables

Step 5: Use differentiation to find the related rate

294.2ft /s

Formal Answer

When the radius is 5 feet, the area is changing at a rate of …

Steps for Word Problems

1. Draw a diagram

2. Define the variables

3. Write down all the information in terms of the variables defined

4. Set up a relation between the variables

5. Use differentiation to find the related rate. Formally answer the question.

Remark on Classwork #2

To save time, problem number 2 does not required all the steps.

Expectations

Use equal signs correctly. Use and notations correctly. Pay attention to the independent

variables: Is it or ? Pay attention to the units.