The derivative of a circle's area

Question

If we take the formula $A=\pi r^{2}$ and differentiate both sides with respect to $t$, we get that $\frac{dA}{dt}=2\pi r\cdot\frac{dr}{dt}$. Now if we increase the radius by $3$, we get the formula $\frac{dA}{dt}=2\pi r\cdot3=6\pi r$. Thus we should expect that the difference in area of a circle with radius $2$ when increased by $3$ (radius will equal $5$) should be $12\pi$. However, this is incorrect; what went wrong?

Answer 1

The derivative at $r=2$ gives the rate of change in a neighborhood of $r=2$, thus a good estimation of the change of the area $A$ can be obtained for $r\approx 2$.