Question:
The side of a square sheet is increasing at the rate of $4 \mathrm{~cm}$ per minute. At what rate is the area increasing when the side is $8 \mathrm{~cm}$ long?
Solution:
Given : $A=x^{2}$ and $\frac{d x}{d t}=4 \mathrm{~cm} / \mathrm{min}$
Let $x$ be the side of the square and $A$ be its area at any time $t .$ Then,
$A=x^{2}$
$\Rightarrow \frac{d A}{d t}=2 x \frac{d x}{d t}$
$\Rightarrow \frac{d A}{d t}=2 \times 8 \times 4$ $\left[\because x=8 \mathrm{~cm}\right.$ and $\left.\frac{d x}{d t}=4 \mathrm{~cm} / \min \right]$
$\Rightarrow \frac{d A}{d t}=64 \mathrm{~cm}^{2} / \min$