12.6.5. Related Time Rates

A particle is moving along the curve . If the rate of change of its -coordinate with respect to time as it passes through the point is 2, find the rate of change of its -coordinate with respect to time at the same point.

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Example

A particle is moving along the curve . If the rate of change of its -coordinate with respect to time as it passes through the point is 2, find the rate of change of its -coordinate with respect to time at the same point.

Solution

In the equation , which describes a particle's movement, is a function of , and is a function of .

First, keep this in mind, and use implicit differentiation as follows:

It is given that the rate of change of the particle's -coordinate with respect to time as it passes through the point is 2. For this reason, now substitute for , 3 for , and 2 for in the equation , and solve for .

Thus, the rate of change of the particle's -coordinate with respect to time at the point is .

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