The derivative of spherical volume formula is spherical surface area. What is the meaning of derivative of spherical surface area formula? For example, the derivative of 2 π R is 2 π, which is the circumference angle, but what is the meaning of the derivative of 8 π r of 4 π R2?

The derivative of spherical volume formula is spherical surface area. What is the meaning of derivative of spherical surface area formula? For example, the derivative of 2 π R is 2 π, which is the circumference angle, but what is the meaning of the derivative of 8 π r of 4 π R2?

Formally: the derivative of the volume of the ball = the surface of the ball
Derivative of area of circle = circumference of circle
The derivative of the circumference of a circle = the circumference angle of the whole circle
In the sense: the derivative of the volume of the ball ≠ the surface of the ball
Derivative of area of circle ≠ circumference of circle
The derivative of the circumference of a circle is not equal to the circumference angle of the whole circle
[coincidence in form is only accidental, but difference in meaning is inevitable]
Because the circle is the most special figure:
Circumference of circle = ∑ arc length of small sector
=Σ radius of circle × radian of small sector
=Radius of Σ circle × Δ θ
= R∑Δθ
= 2πR
=∫Rdθ
= 2πR
Area of circle = ∑ circumference of small ring × width of small ring
= ∑2πr×Δr
=∫2πrdr
= πR²
The volume of the ball = ∑ the area of the small shell × the thickness of the small shell
= ∑4πr²×Δr
=∫4πr²dr
= 4πR³/3
These are the basic ideas and methods of integral
That is: divide, sum, take limit (transition to integral)
Derivative is the rate of spatial change
If the radius of the sphere is changing, the meaning of the derivation of the radius is:
[change of sphere volume caused by one unit of radius change]
It is exactly equal to the surface area of the ball in the magnitude
The explanation of area and perimeter of circle is completely similar
This is a coincidence. It doesn't hold true for ellipses (balls), triangles, squares, cubes,..., etc!
As an interesting category, OK;
As a method classification, Na = not applicable
Circle is just special Case.Not common sense.