Concept preview

What you will be able to do

1. 01Recognise a related-rates situation: two or more quantities change together over time and a relation between them links their rates $\dfrac{dx}{dt}$, $\dfrac{dy}{dt}$, $\dfrac{dA}{dt}$, etc.
2. 02Carry out the related-rates SETUP in order — identify the variables and known/unknown rates, write the geometric or defining relation, differentiate both sides with respect to $t$ using the chain rule, and only THEN substitute the instant's values.
3. 03Apply the setup to the classic relations — an expanding circle $A=\pi r^2$, a growing sphere $V=\tfrac{4}{3}\pi r^3$, and a sliding ladder $x^2+y^2=25$ — and avoid the two fatal errors of substituting numbers too early or using the wrong relation.

A related-rates problem always has the same shape. Several quantities vary as time passes — think of a circular ripple whose radius $r$ and area $A$ both grow, or a ladder whose foot-distance $x$ and top-height $y$ both change as it slips. A fixed relation always ties them: for the ripple $A=\pi r^2$, for the ladder $x^2+y^2=25$. We are told one rate (how fast the radius grows, or how fast the foot slides) and asked for another (how fast the area grows, or how fast the top drops) at one particular instant. Nothing is standing still — that is why they are called related RATES.