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Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

We all know that the volume of a sphere is:

$V = \frac{4}{3}\pi r^{3}$

and its surface area is

$S = 4 \pi r^2$

Now we see that

$\frac{dV}{dr} = S$

As well, the area of a circle is

$A = \pi r^2$

The circumference is

$C = 2 \pi r$

Now we see again that

$\frac{dA}{dr}=C$

There may be more that I have not noticed. Why does this relationship occur?

  • 1
    Imagine the area between circles of radius $r$ and $r+\epsilon$.2012-04-08
  • 0
    It is worth to look at http://math.stackexchange.com/questions/67039/why-does-volume-go-to-zero2012-04-08

1 Answers 1