Multiple Dimensions - Starting Out With 2

The area of the circle is given by, $f(r)\quad =\quad \pi { r }^{ 2 }\\ thus,\quad f'(r)\quad =\quad \frac { d(\pi { r }^{ 2 }) }{ dr } \quad =\quad 2\pi r\\ which\quad is\quad nothing\quad but\quad the\quad circumference\quad and\quad hence\quad the\quad perimeter.$

To prove this without actually differentiate it, we can use visualization that the picture gave. The $f'(r)$ represents the change of the area of the circle at certain radius, $r$ . Then, by slightly increasing the radius of the circle, you add a slight area that was just outside of the current area of the circle, as the shaded picture above shows. Then, if the increase of the radius, or $dr$ , gets smaller and smaller, the added area is infinitesimal, so that it is basically the perimeter of a circle.

Area of circle= pie * r^2.. When you differentiate it you'll get 2 * pie * r which is nothing but circumference or perimeter of circle.

<3 its perimeter.