Related Rates

The central idea

Related-rates problems involve quantities linked by an equation and changing with a common time variable. If $F(x(t),y(t))=0$, differentiation with respect to $t$ produces terms such as $F_x dx/dt+F_y dy/dt$. Rates are evaluated at an instant; the geometric relation must remain true throughout the motion.

Definitions, hypotheses, and notation

The sign convention should be chosen before algebra. A distance increasing can be assigned a positive rate and a height falling a negative rate; the differentiated constraint then relates them automatically. If the question asks 'how fast' as a speed, report the magnitude after first finding and interpreting the signed derivative.

Geometry often supplies an intermediate quantity not mentioned in the final question. In the ladder problem, the missing height is found from the Pythagorean relation at the snapshot. In cone or shadow problems, similar triangles may be needed before differentiating. Eliminating extra variables early usually produces one constraint with exactly one unknown rate at the end.

Conceptual meaning

The chain rule transmits change through a constraint. A quantity can be fixed at the instant while still having a nonzero rate, so numerical values should be substituted after differentiating, not before.

A dependable method and decision rule

1. Draw and label a diagram; declare which quantities depend on time.
2. Write one equation relating the changing quantities.
3. Differentiate the entire equation with respect to time.
4. Substitute the instantaneous values and all known signed rates.
5. Solve for the requested rate and interpret its sign and units.

Fully worked example

Common mistakes and why they fail