<s>[INST] Why is the derivative of circle area the circumference? Why is the derivative of the volume of a sphere the surface area? And why is the derivative of the area of a circle the circumference? Too much of a coincidence, there has to be a reason! 
Also, why is the sum of the cubes the square of the sum of the integers?
 [/INST] I'll answer the last question first because that's the easiest one in my opinion.

Prove that
$$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
We will prove this using induction, by first confirming the identity when $n = 1$ and then proving the inductive step.
Base Case: $n = 1$
$$1^3 = \left(\frac{1(1+1)}{2}\right)^2 = \left(\frac{2}{2}\right)^2 = 1^2 = 1$$
Inductive Step:
Suppose the equality is true for some $x$. That is, $$1^3 + 2^3 + 3^3 + \cdots + x^3 = \left(\frac{x(x+1)}{2}\right)^2$$
Now we add $(x+1)^3$ to both sides and get
$$1^3 + 2^3 + 3^3 + \cdots + x^3 + (x+1)^3 = \left(\frac{x(x+1)}{2}\right)^2 + (x+1)^3 = (x+1)^2 \left(\frac{x^2}{4} + x + 1\right) = \frac{(x+1)^2}{4} (x^2 + 4x + 4) = \frac{(x+1)^2 (x+2)^2}{4} = \left(\frac{(x+1)(x+2)}{2}\right)^2$$
Therefore, if the equation is true for some $x$, then it is also true for $x+1$. And because it is true for $1$, it will be true for $2$, for $3$, for $4$, etc.

Prove that the derivative of the area of the circle is the circumference.

We note that the area of the bigger circle is $\pi (r+dr)^2$ and that of the smaller circle is $\pi r^2$. The difference in area, therefore, is $\pi (r+dr)^2 - \pi r^2 = \pi (2rdr + dr^2)$
We can then imagine turning the small "outline" (the area between the large circle and the small circle) into a thin, narrow strip. Assuming that $dr$ is small enough, the strip will be almost a rectangle with area $\pi (2rdr + dr^2)$. Because the height of the rectangle is $dr$, we divide the area by the height to get the width, which is $$\frac{\pi (2rdr + dr^2)}{dr} = \pi (2r + dr)$$
And finally, because $dr$ approaches $0$ as the strip gets thinner, we see that the width approaches $2\pi r$, the circumference.
Similar logic applies to the sphere and the surface area.
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