I have recently realized that the first derivative of the area of a circle $A = \pi r^2$ is the circumference (perimeter) $C = 2 \pi r$ and the first derivative of the volume of a sphere $V = 4 \pi r^3 / 3$ is the surface area of a sphere $SA = 4 \pi r^2$. (So basically 3D to 2D, 2D to 1D).
Then I tried with cube and square and found out that the first derivative of the area of a square $A = a^2$ is half the perimeter of a square, and the first derivative of the volume of a cube $V = a^3$ is half the surface area of a cube.
Is there any mathematical explantion to this?
Best Answer
For the sphere and circle there are wonderful explanations. Think of it this way - when the volume of a sphere increases, it is like you are adding infinitely many infinitely thin (flat) layers (or shells) to the outside of the sphere, each with an area of $4\pi r^2$.
Now imagine, when you add area to a square, you are adding a corner in a v-shape to the square with a length of half of the perimeter. The 3D equivalent of this is adding a pyramid-like shape covering three mutually adjacent faces of the cube to the corner of a cube each time the volume increases.
You can even think about it in one dimension. If the length of a line segment is given by $d$, then its first derivative is $1$, and you are adding infinitely many points each time the length increases.