The problem statement:
Oil spreads on a frying pan so that it's radius is proportional to $t^{1/2}$, where $t$ represents the time from the moment when the oil is poured. Find the change $dT/dt$ of the thickness T of the oil.
From the solution I can tell that we should assume that the oil is in the shape of a cylinder and that the volume is constant.
Implicit solution (according to textbook):
$r = kt^{1/2}$
$r^2T = \frac{V}{\pi} = c$ (definition of the volume of a cylinder)
$2r\frac{dr}{dt}T + r^2\frac{dT}{dt} = 0$
$\frac{dT}{dt} = -\frac{2}{r}\frac{dr}{dt}$
$\frac{dT}{dt} = -\frac{1}{t}$ (after computing the derivative of r)
According to the textbook this is the correct solution, but if we differentiate explicitly:
$r = kt^{1/2}$
$T = \frac{V}{r^2\pi} = \frac{V}{k^2t\pi}$
$\frac{dT}{dt} = -\frac{V}{k^2\pi} \frac{1}{t^2} = -\frac{r^2\pi T}{k^2\pi} \frac{1}{t^2} = -\frac{k^2tT}{k^2} \frac{1}{t^2} = -\frac{T}{t}$
If both of the solution are correct then $\frac{dT}{dt} = -\frac{1}{t} = -\frac{T}{t}$ and $T = 1$, which can't be possible. Why does differentiating implicitly and explicitly give different contradictory solutions?
When you move from the third to the fourth line in the book solution, you drop a factor of $T$. The fourth line should read $\frac{dT}{dt}=-\frac{2T}{r}\frac{dr}{dt}$.