Consider everything a function of time (t). For your circle:
We find the derivative with respect to t = time.
On the left-had side, we have simply .
On the right-hand side, we have
Putting it all together (and dispensing with the cumbersome '(t)' notation), we get
Knowing this, one is armed with enough to solve many problem types.
Actually, we may also want to dispose of the cumbersome 'dt' notation and just go with the differential usage.
In words, this says, "The change in the area is multiplied by the change in the radius.
We are given dr = 15 cm/s.
Let's see what you get.