TITLE: Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
QUESTION [115 upvotes]: When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi r^2$.
Is this just a coincidence, or is there some deep explanation for why we should expect this?

REPLY [5 votes]: I recommend the article by J. Tong, Area and perimeter, volume and surface area, College Math. J. 28 (1) (1997) 57. He shows that for any region where the area can be written as $A(s)=c s^2$ and the perimeter as $L(s)= k s$, you can set $x=(2c/k) s$, and you will get $A'(x)=L(x)$. That means that by careful parametrization, the above holds for rectangles and ellipses, too.