Derivative of the Area of a Circle

[NFUN]

It's the perimeter.

Power Rule: d/dx[xⁿ]=nx^n-1

Constant Rule: d/dx[nx]=n*d/dx(x).

A=(pi)r²

d/dr[(pi)r²]=2r(pi)

P=2r(pi)

I have nothing else to say, and I'm sure most of those who know what derivatives are are aware of this, but I just wanted to put this out there.

That is all, good day.

Edited October 11, 2014 by NFUN

[ZetaX]

The reason:

Pick some small number s. The derivative measures the increase of the area if the radius grows by a small number; thus the difference between the areas of a circle with radius r+s and one of radius r is approximately s times the derivative.

On the other hand, the difference is just the area of a ring with inner radius r and thickness s. That ring has approximately an area of s times the perimeter of the circle.

In total: s·derivative = s·perimeter. Thus derivative=perimeter.

Note that nowhere you need to actually know a formula for the perimeter or the area of a circle.

To make it formal, use the inequality s·permieter_1 < area_of_the_ring < s·perimeter_2, where the two perimeters are the inner and outer one, respectively. Then take the limit of s going to 0.

This is by the same argument also true for spheres: the derivative of the volume is the surface area (add "hyper-" for higher dimensional analogues).

[metaphor]

I think this also works for any radially symmetric shape, as long as you express the formulas and take the derivative with respect to a distance from the center that's perpendicular to a side. For example, for a square of side length a, let r = a/2. Then its perimeter is 8r while its area is 4r^2.

[Idobox]

It also works on spheres

Volume = 4/3 * pi * r^3

Area = dV/dr = 4* pi * r²