Objectives:1. Be able to find the derivative of an equation with

respect to various variables.2. Be able to solve various rates of change applications.

Critical Vocabulary:Derivative, Rate of Change

I. Derivatives

Example 1: Find the derivative of y with respect to x: x2 + y2 = 25

022 dx

dyyx x

dx

dyy 22

y

x

dx

dy

2

2

y

x

dx

dy

Example 2: Find the derivative of x with respect to y: x2 + y2 = 25

022 ydy

dxx y

dy

dxx 22

x

y

dy

dx

2

2

x

y

dy

dx

I. Derivatives

Example 3: Find the derivative of y with respect to t: x2 + y2 = 25

022 dt

dyy

dt

dxx

dt

dxx

dt

dyy 22

ydt

dxx

dt

dy

2

2

ydtdxx

dt

dy

Example 4: Find the derivative of x with respect to t: x2 + y2 = 25

022 dt

dyy

dt

dxx

dt

dyy

dt

dxx 22

xdt

dyy

dt

dx

2

2

xdtdyy

dt

dx

I. Derivatives

Example 5: Find dy/dt when x = 2 of the equation 4xy = 12 given that dx/dt = 4

xxf 4)( dt

dxxf 4)('

yxg )(dt

dyxg )('

044 dt

dyx

dt

dxy

dt

dxy

dt

dyx 44

xdtdxy

dt

dy

(If x = 2, Then y = 3/2)

2

423

dt

dy

2

6

dt

dy

3dt

dy

Page 304 #1-7 odd

Objectives:1. Be able to find the derivative of an equation with

respect to various variables.2. Be able to solve various rates of change applications.

Critical Vocabulary:Derivative, Rate of Change

WARM UP: Find dy/dt: 3x2y3 = 12

I. Derivatives

Warm Up: Find dy/dt: 3x2y3 = 1223)( xxg

dt

dxxxg 6)(

3)( yxh dt

dyyxh 23)(

096 223 dt

dyyx

dt

dxxy

dt

dxxy

dt

dyyx 322 69

22

3

9

6

yxdtdx

xy

dt

dy

xdt

dxy

dt

dy

3

2

II. Applications

Guidelines for Solving Related Rate Problems

1. Identify all given quantities and quantities to be determined.• Make a sketch and label your diagram

2. Write an equation involving the variables whose rates or change either are given or are to be determined.

• Volume Formulas (Inside Cover of Book)

• Area Formulas (Inside Cover of Book)

• Pythagorean Theorem (when you sketch looks like a RT Δ)

3. Using implicit Differentiation, differentiate with respect to time.

4. Substitute Values as necessary. Then solve for the required rate of change.

II. Applications

Example 6: A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (r) of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area (A) of the disturbed water changing?

What do I know: sec/1 ftdt

dr feetr 4 2rA

What do I need to find:dt

dA

2rA

dt

drr

dt

dA 2

Differentiate:

Substitute: )1)(4(2dt

dA

8dt

dA

The total area of the disturbed water is

changing at 25.13 ft2/sec.

II. Applications

Example 7: Air is being pumped into a spherical at a rate of 4.5 cubic feet per minute. Find the rate of change of the radius when the radius is 2 feet.

What do I know: min/5.4 3ftdt

dV feetr 2 3

3

4rV

What do I need to find:dt

dr

3

3

4rV

dt

drr

dt

dv 24

Differentiate:

Substitute:2)2(4

2/9

dt

dr

The rate of change of the radius when the radius is 2 feet is 0.09

ft/min.

24

/

r

dtdV

dt

dr

329

dt

dr

Page 304-306 #13-23 odd

Objectives:1. Be able to find the derivative of an equation with

respect to various variables.2. Be able to solve various rates of change applications.

Critical Vocabulary:Derivative, Rate of Change

II. Applications

Example 8: A ladder 10 feet length is leaning against a brick wall. The top of the ladder is originally 8.5 feet high. The top of the ladder falls at a fixed rate of speed dy/dt. As time goes by the distance x(t) from the base of the wall to the bottom of the ladder changes. What is the rate of change of the distance x(t)?

II. Applications

Example 8: A ladder 10 feet length is leaning against a brick wall. The top of the ladder is originally 8.5 feet high. The top of the ladder falls at a fixed rate of speed dy/dt. As time goes by the distance x(t) from the base of the wall to the bottom of the ladder changes. What is the rate of change of the distance x(t)?

What do I know: 222 zyx feety 5.8 feetx 27.5

What do I need to find:dt

dx

10022 yx

022 dt

dyy

dt

dxx

Differentiate:

Substitute:

1dt

dyfeetz 10

dt

dyy

dt

dxx 22

x

dtdyy

dt

dx /

27.5

)1()5.8(

dt

dx

61.1dt

dx

II. Applications

Example 9: A baseball diamond has the shape of a square with sides 90 feet long. Suppose a player is running from 1st to 2nd at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base.

II. Applications

Example 9: A baseball diamond has the shape of a square with sides 90 feet long. Suppose a player is running from 1st to 2nd at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base.

What do I know: 222 zyx feety 60 feetx 90

What do I need to find:dt

dz

228100 zy

dt

dzz

dt

dyy 22

Differentiate:

Substitute:

sec/28 ftdt

dy feetz 17.108

z

dtdyy

dt

dz /

17.108

)28()60(

dt

dz

sec/53.15 ftdt

dz

1. Page 304-306 #27, 332. Worksheet: “Applications: Rate of Change”