If two quantities are related , how does the derivative of one depend on the other?

For example, we know that the volume of a spherical balloon depends on the radius according to

V = 4/3 (pi) r³

If r=r(t), then dV/dt = dV/dr dr/dt = 4 (pi) r² dr/dt. Another way to express this is to say that dr/dt = 1/(4 (pi) r²) dV/dt. Given one rate of change, and the values of r at a given time t, we can find the other rate of change.

Another example is the "falling ladder" problem. A ladder 10 feet tall rests against a wall. If the bottom of the ladder is pulled away at 1 ft per second, how fast is the top sliding down when the base is 6 feet away?

Using the Pythagorean theorem, x² + y² = 10 ² = 100, we have (by implicit differentiation and the chain rule)

2 x(t) dx/dt + 2 y(t) dy/dt = 0

Therefore, dy/dt = -x/y dx/dt. When x=6, y=8. Since dx/dt = 1, we have dy/dt = -6/8 ft/sec. Notice, when y=0, dy/dt=infinity! Is this reasonable?

Strategy

1. Read the problem carefully!
2. Draw a diagram (if possible)
3. Assign symbols to all quantities that a functions of time
4. Express information in terms of derivatives
5. Write an equation relating the variables of the problem
6. Use chain rule (implicit differentiation) to relate derivatives
7. Solve for the unknown rate

Last modified Mon Oct 14 19:27:30 CDT 1996