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

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

I. Derivatives

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

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

I. Derivatives

Example 5: Find dy/dt when x = 2 of the equation 4xy = 12 given that dx/dt = 4(If x = 2, Then y = 3/2)

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 = 12

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?

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.

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 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.