Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies.

Let T be the total surface area of a cylinder. Then,

$T=2 \pi r(r+h)$

Since the radius varies, we differentiate the total surface area w.r.t. radius r.

Now,

$\frac{d T}{d r}=\frac{d}{d r}[2 \pi r(r+h)]$

$\Rightarrow \frac{d T}{d r}=\frac{d}{d r}\left(2 \pi r^{2}\right)+\frac{d}{d r}(2 \pi r h)$

$\Rightarrow \frac{d T}{d r}=4 \pi r+2 \pi h$

$\Rightarrow \frac{d T}{d r}=2 \pi(r+h)$