Let A be the area (the dependent variable) and r be the radius (the independent variable). As the radius changes from r to r + dr, the area changes from πr^{2} to π(r + dr)^{2}. Therefore,

dA = π(r + dr)^{2} − πr^{2}
= π[r^{2} + 2r dr + (dr)^{2}] − πr^{2}
= 2πr dr + π (dr)^{2}.

The π (dr)^{2} is called a “second order” term; we can neglect it because it’s very small (smaller than dr itself). Thus

dA = 2πr dr,

or

\frac{dA}{dr} = 2πr.