Implicit Differentiation/Related Rates


Common Errors!!

  • Error 1 - labeling pieces that change over time with a constant value
  • Error 2 - labeling pieces that are changing but which are irrelevant to the problem. Variable names are assigned to pieces that are changing if and only if you are given their rate of change or you are explicitly asked to find their rate of change.
  • Error 3-assigning positive derivative values to pieces that are decreasing over time

Find the formula for the first derivative of the function two ways.

1.Use Explicit Differentiation. (i.e. differentiate the explicitly stated function.)

2.Use Implicit Differentiation. (i.e., differentiate the implicit equation )


Find the slope of the curve in Figure 2 at the point .

1)Differentiate both sides of the equation with respect to x.

  • Remember to treat y as a function of x and use the chain rule. For example:

2)Algebraically solve for .

3)Replace x and y with their values.

Figure 2:

Find the slope of the curve in Figure 3 at the point .

1)Differentiate both sides of the equation with respect to x.

  • Remember to treat y as a function of x and use the chain rule. For example:

2)Algebraically solve for .

3)Replace x and y with their values.

Figure 3:

Solve each Related Rates problem.

In Figure 4 x,y, and z represent the lengths of the indicated sides measured in inches. The 3 lengths are changing with respect to time, t, measured in seconds. The side labeled yis increasing at the constant rate of 4 in/s and the side labeled x is decreasing at the constant rate of 3 in/s. Find the rate at which the side labeled z is changing at the instant and . What is the instantaneous rate of change in zat the instant and ? In each case make sure that you clearly communicate whether zis increasing or decreasing.

In Figure 5 x represents the length of the indicated side (cm) and represents the measurement of the indicated angle (rad); both variables are changing with respect to time, t, measured in seconds. The lengths of the other two sides are fixed at 4 cm and 7 cm. is increasing at a constant rate of 4 deg/min. Find the rate at which x is changing at the instant. Make sure that you clearly communicate whether xis increasing or decreasing.

In Figure 6 xandyrepresent the lengths of the indicated sides measured in feet. The 2 lengths are changing with respect to time, t, measured in seconds. The side labeled yis increasing at the constant rate of .2 ft/s. Find the rate at which the side labeled x is changing at the instant. Make sure that you clearly communicate whether xis increasing or decreasing.

At noon one day a truck is 250 miles due east of a car. The truck is travelling west at a constant speed of 25 mph and the car is travelling due north at a constant speed of 50 mph. At what rate is the distance between the two vehicles changing 15 minutes after noon?

A light house sits on an island of the coast of Maine. The beam source is exactly two miles from the coast along the perpendicular line from the beam source to the coast. The beam makes one complete revolution every 10 seconds. Find the rate at which the beam moves along the coast at the instant the angle between the beam and the perpendicular line to the coast is 50.

To solve this problem you must assume that the coastline is perfectly straight. You must also assume that the light beam is laser like and at ground level.

A tank filled with water is in the shape of an inverted cone 20 feet high with a circular base (on top) whose radius is 5 feet. Water is running out the bottom of the tank at the constant rate of 2 ft3/min. How fast is the water level falling when the water is 8 feet deep?

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