A ladder slides down a wall — how fast is the top falling when I know how fast the base moves? (related rates)
The ladder length L is fixed, so x² + y² = L², where x is the base distance and y the height.
The ladder length L is fixed, so x² + y² = L², where x is the base distance and y the height. Differentiate both sides with respect to time: 2x(dx/dt) + 2y(dy/dt) = 0, so dy/dt = −(x/y)(dx/dt). Plug in the known base speed dx/dt and the current x, y to get how fast the top drops. Related-rates problems all work this way: link the variables by an equation, then differentiate with respect to time.
Key point
Relate the variables by an equation, differentiate w.r.t. time, then substitute the known rate.
Common mistake
differentiating with respect to x instead of time t.
Memory tip
related rates: one equation linking the variables, then d/dt everything.
Tap to watch or see it right here — the app stays open (it only opens a quick search if nothing embeddable is found).