why is the surface area of a cube not the derivative of a volume?
You have to start from the origin. Try calculating it in terms of r, where x=2r. You will get the right answer
Nevermind, I lied.
Ok, I was half right.
The formula works for cubes and squares if you think of the area of a square being 4r^2, and the volume of cube being 8r^3, where r is the radius and x=2r
The surface area of a cube is not the derivative of its volume because these two quantities are not related through differentiation.
To understand why, let's first briefly discuss what each of these terms represents. The volume of a cube refers to the amount of space enclosed within its boundaries, whereas the surface area refers to the total area of all its faces.
The volume of a cube is given by the formula V = s^3, where s represents the length of one side of the cube. Taking the derivative of this formula with respect to s would give us the rate at which the volume is changing with respect to changes in s, which represents how sensitive the volume is to changes in the side length.
On the other hand, the surface area of a cube can be calculated using the formula A = 6s^2, where A represents the surface area and s is the side length. Taking the derivative of this formula with respect to s would give us the rate at which the surface area is changing with respect to changes in s, representing how sensitive the surface area is to changes in the side length.
So, while both the volume and surface area of a cube are functions of the side length, they are distinct quantities and their relationship is not captured by differentiation. The surface area is not the derivative of the volume because the derivatives of these two formulas do not give similar results.
In summary, the surface area and volume of a cube are related, but their relationship cannot be explained by differentiation.