I don't know whether this a rigorous mathematical question. But I was trying to figure out the intuition behind the answer . I have through out my high school used the fact that d/dr (πr^2)= 2πr . It is obvious from the formula. My question is how should I understand the answer intuitively or geometrically ?
We all know that the rate of change of displacement with time is the instantaneous velocity. The answer seems obvious in the first reading of any basic calculus text.
Derivatives of certain other functions also seem obvious (or atleast not as unobvious as the above circle one ) at the first go , or if they don't seem so obvious we can understand them after drawing a picture and analysing it geometrically.
But how should I intuitively understand the rate of change of area of circle with radius being equal to the circumference of the circle ? Why is it so ? I need to understand it geometrically too. How can the rate of change of area of circle with radius give the circumference of the circle ( intuitively ) ?