Q.The length of a rectangle is increasing at the rate of $3\ \text{cm/s}$ and its width is increasing at the rate of $4\ \text{cm/s}$. Find the rate of increase of the area of the rectangle at the instant when the length is $12\ \text{cm}$ and the width is $5\ \text{cm}$.

Many quantities in geometry and physics are functions of time (or of some other varying quantity), and the derivative measures how fast one quantity changes as another changes. If $y=f(x)$ and both $x$ and $y$ vary with time $t$, then $dy/dt=(dy/dx)\cdot(dx/dt)$ by the chain rule — this lets us relate the rate of change of $y$ to the rate of change of $x$ once we know how $y$ depends on $x$. Typical situations combine a geometric formula (area of a circle, volume of a sphere or cone, area of a triangle, etc.) with a given rate for one variable to find the rate of another variable at a specific instant. For problems involving similar triangles (a lamp-post and a walking person's shadow, a ladder sliding down a wall), first set up an algebraic relation between the two changing lengths using similar triangles or the Pythagorean theorem, then differentiate that relation wi …