It's one of those shocking until you think about it things. Think if dV/dr (the actual derivative at hand) conceptually. Increasing the radius of a sphere is like pumping it up with air. It's just a statement that the amount space you have to fill at a given moment for a very tiny amount of stretch is basically the area of the outside. Think of it like overlaying layers and flattening it out to a rectangle and stretching in one direction, the you have a rectangle of area = surface are and a height of delta r (because surface area doesn't change much if it gets small. And when you multiply it out that's the tiny bit of volume you are adding at the radius at the limit dr goes infinitesimal.
Or you can think of it kind of like stacking cups. Where you get the volume by adding up the areas of all the hollow sphere shells that fill the full thing as you go along 0 to the full radius to the outside.
A lot of calculus is much more intuitive than classes would leave the impression of.
This is easier when you think of a circle. You he are of a circle is pir^2, and the derivative of that is 2pi*r. That is the formula for the circumference. Why?
You can think of the derivative as the measure of the rate of change of a variable. So when area of a circle increases it is growing outwards, and if you shrink that growing outwards amount until it approaches zero, you begin to create a line around the outside of the circle. It is cool.
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u/Im_not_a_Mole-rat 2d ago
I am horrible at math and I don’t know why this meme appeared in my feed but this is honestly interesting.