Answer
The rate of change of the area of the circle is equal to the circumference of the circle.
Work Step by Step
The area of a circle is given by the formula $f\left( x \right)=\pi {{x}^{2}}$, where x is the radius of the circle.
The rate of change is given by the formula, ${f}'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( x+h \right)-f\left( x \right)}{h}$
Find the rate of change of the area of the circle by the above formula,
$\begin{align}
& {f}'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{\pi {{\left( x+h \right)}^{2}}-\pi {{x}^{2}}}{h} \\
& =\underset{h\to 0}{\mathop{\lim }}\,\frac{\pi \left( {{x}^{2}}+{{h}^{2}}+2xh \right)-\pi {{x}^{2}}}{h} \\
& =\underset{h\to 0}{\mathop{\lim }}\,\frac{\pi {{x}^{2}}+\pi {{h}^{2}}+2\pi xh-\pi {{x}^{2}}}{h} \\
& =\underset{h\to 0}{\mathop{\lim }}\,\frac{\pi {{h}^{2}}+2\pi xh}{h}
\end{align}$
On further simplification
$\begin{align}
& {f}'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\left( \pi h+2\pi x \right) \\
& =\pi \left( 0 \right)+2\pi x \\
& =2\pi x
\end{align}$
It is known that the circumference of the circle is $2\pi x$, where x is the radius of the circle.
Therefore, the rate of change of the area of the circle is equal to the circumference of the circle.
