I have a very simple question, but different methods lead me to different solutions which is where I am confused.
(Q) Obtain a relationship for change between circumference $C$ and the area $A$ of a circle over time.
$ A = \pi r^2 \\ C = 2\pi r $
Method 1:
So $A = \frac{1}{4\pi}C^2$ after substituting $r$ in terms of $C$. Then implicit differentiation gives us $\frac{dA}{dt} = \frac{1}{2\pi} \cdot \frac{dC}{dt}$
Method 2:
Differentiating the first two equations, we have $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ and $\frac{dC}{dt} = 2 \pi \frac{dr}{dt}$ respectively. Then I isolated for $\frac{dr}{dt}$ in the second equation here and substituted into the first equation resulting in the relation $\frac{dA}{dt} = 2\pi r \left(\frac{1}{2\pi} \cdot \frac{dC}{dt}\right) = r \frac{dC}{dt}$.
I believe the second method is incorrect; however, I am unsure why it is incorrect. If there is any reading that might solidify my knowledge in terms of related rates/implicit differentiation, I would appreciate it.