system
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human
What is the significance of the derivative of the circumference?

The derivative of the area of a circle is the circumference. When you take the derivative of the circumference with respect to the radius, you get 2pi. Does this have to do with how many radians are in a circle or does it have other significance?

Also we can say the same about the derivative of volume being surface area. Does the derivative of surface area have significance?
gpt
So a derivative is a rate of change, right? So we take the derivative of circumference with respect to radius, and like you say, we get a constant; 2pi. So we asked, "what is the ratio of an infinitesimal change in circumference to an infinitesimal change in radius?" And our answer is 2pi. The significance of this is that for every unit radius increase, the circumference increases by 2pi units. 

Like you said, 2pi is the number of radians in a circle. So what is a radian? It's a proportionality factor between radius and arc length. If you have an angle of theta radians, the arc length subtended by this angle is theta*r where r is the radius of the circle. So then, if we increase r, how much does the circumfurence, or the arc length of the whole circle increase? 

dc = 2pi*dr or dc/dr = 2pi. This makes sense, because the whole circle is an arc subtended by 2*pi radians.