2. Implicit Differentiation
Not all functions can be easily written in a form where the
independent variable is completely isolated from the dependent
variable, and for some relations it is simply not possible. Functions
and relations of these types are called implicit. Equations like
𝑥3 + 𝑦3 + 4𝑥𝑦 = 0, 𝑦2 − 2𝑥 = 0, 𝑥2 + 𝑦2 − 36 = 0
define an implicit relation between the variables x and y, meaning
that a value x determines one or more values of y, even though we
do not have a simple formula for the y values.
3. Implicit Differentiation
• STEPS IN PERFORMING 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.
9. Related Rates
• STEPS IN SOLVING PROBLEMS INVOLVING RELATED RATES
1. Read and understand the problem.
2. Make a carefully labeled diagram.
3. Write down and label constants, variables, rates and what is being
sought.
4. Write a function that relates the variables.
5. Differentiate all terms with respect to time. 6. Substitute known
quantities.
10. Examples
A 20–foot ladder rests against a vertical wall. If the bottom of the
ladder is sliding away from the base of the wall at the rate of 2 ft/sec,
how fast is the top of the ladder moving down the wall when the
bottom of the ladder is 6 feet from the base?
11. Examples
Air is being pumped into a spherical balloon at a rate of 4 cubic
inches per minute. Find the rate of change when the radius is 6
inches.
12. Examples
• A zero-depth pool is angled downward at 23˚. You are walking
steadily toward the deeper water at a rate of 3 feet per second. At
the instant you are 14 feet from the edge of the water, how fast is
the water level rising on you?
13. Examples
Water runs into a conical tank at the rate of 10 𝑓𝑡3/min . The tank
stands point down and has a height of 12 ft and a base radius of 6 ft.
How fast is the water level rising when the water is 8ft deep?
14. Examples
A water droplet falls on a still pond and creates a concentric circular
ripple that propagates away from the center. Assuming that the area
of the ripple is increasing at the rate of 2𝜋
𝑐𝑚2
𝑠
, find the rate at which
the radius is increasing at the instant when the radius is 10 cm.
