4.4 Introduction to Related Rates

What are related rates?

Related rates problems involve two or more quantities that change with respect to time and are connected by an equation. By differentiating the equation implicitly with respect to time (t), you relate the unknown rate of change to known rates. The key is finding a formula linking the variables before differentiating.

Problem-solving steps

1) Draw a diagram and label variables. 2) Write an equation relating the variables. 3) Differentiate both sides with respect to t, using the chain rule for each variable. 4) Substitute known values (rates and quantities) at the specific instant. 5) Solve for the unknown rate. Always substitute numbers *after* differentiating, not before.

Common setups

Classic examples include a ladder sliding down a wall (Pythagorean theorem), a balloon inflating (volume of a sphere), and shadows or cones filling with liquid (similar triangles or volume formulas). Each requires identifying which rate is given (dx/dt) and which is sought (dy/dt), then linking them through geometry.

Key terms

Related rates

Problems involving two or more variables that change with time, connected by an equation, solved via implicit differentiation with respect to t.

Steps to solve related rates

1) Diagram/label 2) Write equation 3) Differentiate w.r.t. t 4) Substitute known values 5) Solve for unknown rate.

Why use the chain rule?

Because each variable is itself a function of time, so differentiating requires multiplying by its own rate (dx/dt, dy/dt, etc.).

Common mistake in related rates

Substituting numerical values before differentiating, which incorrectly treats variables as constants.

Example: Sphere volume rate

For V = (4/3)πr³, differentiate to get dV/dt = 4πr²(dr/dt), relating volume and radius rates of change.

Try an AP-style question

x and y depend on time t, and x² + y² = 25. Differentiating with respect to t gives…

1. x(dx/dt) + y(dy/dt) = 25
2. 2x + 2y = 0
3. 2x(dx/dt) + 2y = 0
4. 2x(dx/dt) + 2y(dy/dt) = 0