Related-rates problems are exercises designed to illustrate the use of the chain rule. Varying quantities linked by some quantitative relationship give rise to "related rates of change." The calculus content of the typical such problem is minimal; the challenge in a related-rates problem, which is typically a "word problem," is expressing the relationship in terms of the appropriate variables.

For example, suppose the quantity $y$ depends on the quantity $x$, and that $x$ in turn changes in time, $t$. Of course, that induces a related change in $y$, and thus, this change in $y$ is called a "related rate." In this simply stated example, $y\=y\left(x\left(t\right)\right)$, and the rate of change of $y$, namely, $\frac{\mathrm{dy}}{\mathrm{dt}}$, is given by the chain rule as $\frac{\mathrm{dy}}{\mathrm{dx}}\cdot \frac{\mathrm{dx}}{\mathrm{dt}}$. That's all the calculus involved in the typical related-rate problem. Again, the challenge is extracting the formula $y\left(x\right)$ from the words of the statement of the problem.$$

One last tip: The typical related-rate problem asks for a "related rate" at some specific moment. Almost never is this magic moment given, or need it be found. The time at which the measurement to be made is coincident with the state of one of the variables in the problem, and it is the value of that variable that is used to fix the value of the related rate.  See the Examples 3.6.1-5 for illustrations of this "feature" of most such problems.$$