Answer
$dS=\frac{\pi rh_0}{\sqrt {r^2+h_0^2}}dh$
Work Step by Step
Step 1. Given the formula for the lateral surface $S=\pi r\sqrt {r^2+h^2}$, we keep $r$ constant and get $S'=\pi r\frac{2h}{2\sqrt {r^2+h^2}}=\frac{\pi rh}{\sqrt {r^2+h^2}}$
Step 2. The change in lateral surface area when the height changes from $h_0$ to $h_0+dh$ is given by:
$dS=S'(h_0)dh=\frac{\pi rh_0}{\sqrt {r^2+h_0^2}}dh$
