Example 1
The derivative also represents rate of change. Do the following for related rate problems:
Try it!
The voltage of a certain thermocouple as a function of temperature is given by
E = 2.800T + 0.006T²
If the temperature is increasing at the rate of 0.5° C/min, how fast is the voltage increasing when T = 100°C?
Find: when T=100° C Solution: E = 2.8T + 0.006T² use implicit differentiation = 2.8 + 2 (0.006) T Now substitute what is given = 2.8 (0.5) + 2 (0.006) (100) (0.5) = 1.4 + 0.6 = 2 
Example 2
To solve related rate problems (after translating "given" and "to find"):
Try it!
Problem 1:
A spherical balloon is being blown up at a constant rate of
2 ft³/min. Find the rate at which the radius is increasing when the radius is 3 feet.
Rate means derivative, so translation becomes:
Given: = 2 ft³/min (ft³ means volume) Find: = ? when r = 3 ft 
First we need an equation relating v and r (see inside front cover of text):
1. Equation for volume of sphere:  
2. Differentiate equation (with respect to t)  
3. Substitute in given  
4. Solve for dr/dt 
Problem 2:
A 15 foot ladder is sliding down the side of a vertical building. If the top of the ladder is moving at the rate of .25 ft/min, how fast is the foot of the ladder moving when it is 9 feet from the building?

Given: dx/dt = 0.25 ft/min (negative because the ladder is moving down the wall)

You need to find the value for x before you substitute into this equation:
x² + 9² = 15² x² + 81 = 225 x² = 144 x = 12 
Now since: