AcademyApplications Of Derivatives
Academy
Related Rates
Level 1 - Calculus topic page in Applications Of Derivatives.
Principle
Related rates problems track quantities that change together over time. Equations relate the quantities, and derivatives with respect to time relate their rates.
Notation
\(t\)
time
\(\frac{dx}{dt}\)
rate of change of x with respect to time
\(\frac{dy}{dt}\)
rate of change of y with respect to time
The Core Method
Write an equation relating the variables, differentiate both sides with respect to \(t\), substitute the known values at the instant, and solve for the unknown rate.
Time derivative pattern
\[\frac{d}{dt}(x^2)=2x\frac{dx}{dt}\]
Substitute after differentiating, not before, unless the quantity is constant for all time.
Worked Cases
Question
A circle has radius increasing at \(3\) units per second. How fast is its area changing when \(r=5\)?
Answer
Area is \(A=\pi r^2\). Differentiate with respect to time: \(\frac{dA}{dt}=2\pi r\frac{dr}{dt}\). Substitute \(r=5\) and \(dr/dt=3\): \(dA/dt=2\pi(5)(3)=30\pi\).
Examples
Question
If \(x^2+y^2=25\), \(dx/dt=2\), and \(x=3\), \(y=4\), find \(dy/dt\).
Answer
Differentiate: \(2x\frac{dx}{dt}+2y\frac{dy}{dt}=0\). Substitute: \(2(3)(2)+2(4)\frac{dy}{dt}=0\). This gives \(12+8dy/dt=0\), so \(dy/dt=-3/2\).