Show that the relative error in computing the volume of a sphere,

Let x be the radius of the sphere and y be its volume.

Let $\Delta x$ be the error in the radius and $\Delta V$ be the approximate error in the volume.

$y=\frac{4}{3} \pi x^{3}$

$\Rightarrow \frac{d y}{d x}=4 \pi x^{2}$

$\Rightarrow \Delta y=d y=\frac{d y}{d x} d x=4 \pi x^{2} \times \Delta x$

$\Rightarrow \Delta y=3 \times \frac{4}{3} \pi x^{3} \times \frac{\Delta x}{x}$

$\Rightarrow \Delta y=3 \times y \times \frac{\Delta x}{x}$

$\Rightarrow \frac{\Delta y}{y}=3 \frac{\Delta x}{x}$

Hence proved.