Overview

Related rates problems involve two (or more) variables that change at the same time, possibly at different rates. If we know how the variables are related, and how fast one of them is changing, then we can figure out how fast the other one is changing. This usually involves writing an equation relating the two variables and taking the derivative of the equation with respect to time. Implicit differentiation is often used.

Example: A particle is moving around a circle of radius 5 around the origin. As it passes through the point $(3,4)$, its $x$ position is changing at a rate $\dfrac{dx}{dt}\Big|_{{x=3},\, {y=4}} = 2$. How fast is $y$ changing at that instant? Solution: We know that the equation for the circle is $$x^2 + y^2 = 25.$$ To find a relationship between the rates of change of $x$ and $y$ with respect to time, we can implicitely differentiate the equation above with respect to $t$. $$2 x \frac{dx}{dt} + 2 y \frac{dy}{dt} = 0.$$ We know that when the particle is passing $(3,4)$, then its velocity is $\displaystyle{\frac{dx}{dt}\Big|_{{x=3},\, {y=4}} = 2}$, so we can solve for $\dfrac{dy}{dt}\Big|_{{x=3},\, {y=4}}$. Since $$2(3)(2) + 2(4)\frac{dy}{dt}\Big|_{{x=3},\, {y=4}} = 0,$$ we have $\dfrac{dy}{dt}\Big|_{{x=3},\, {y=4}}= -\frac{12}{8}=-\frac{3}{2}.$