A 13 m ladder leans against a wall. Its base slides away from the wall at 2 m/s. How fast is the top sliding down when the base is 5 m from the wall? (in m/s)
From \(x^{2}+y^{2}=169\) we differentiate: \(2x\,x'+2y\,y'=0\), so \(y'=-\dfrac{x}{y}x'\). If x = 5, then y = 12 and \(y'=-\dfrac{5}{12}\cdot2=-\dfrac{5}{6}\) m/s (the negative sign means it moves down).
In a related rates problem several quantities change with time \(t\) and are linked by an equation. The equation is differentiated with respect to \(t\) (chain rule) and the known values are substituted.