Why is there no T in the simplified derivative? Why is the expression I have circled lacking the variable T? from simplifying the expression to the right of it, it seems T should be multiplied with the numerator.
For context, I have attached the problem below the solutions which I am unsure about.
thanks


 A: The radius $r$ and thickness $T$ are functions of time $t$, so to write these out explicitly, we have
$$r(t) = Ct^{1/2}, \\ r^2(t) T(t) = \frac{V}{\pi}.$$  Then differentiating with respect to $t$ yields
$$2r(t) \frac{dr}{dt} T(t) + r^2(t) \frac{dT}{dt} = 0,$$ which is equivalent to the text.  This implies
$$\frac{dT}{dt} = \frac{1}{r^2(t)} \left( -2r(t)T(t) \frac{dr}{dt} \right) = -\frac{2T(t)}{r(t)} \frac{dr}{dt}.$$
The text is missing the factor $T(t)$.  However, proceeding with the solution, we note that
$$\frac{T(t)}{r(t)} = \frac{V}{\pi} r^{-3}(t) = K t^{-3/2}$$ for some constant $K$ with respect to $t$.  So $$\frac{dT}{dt} \propto - t^{-3/2} t^{-1/2} = -t^{-2};$$ that is to say, the rate of change in the thickness of the oil is inversely proportional to the square of the time elapsed.
The whole calculation could be greatly simplified, however.  Note that the first two equations imply
$$T(t) \propto r^{-2}(t) \propto t^{-1},$$ therefore, $$\frac{dT}{dt} \propto -t^{-2}.$$  It makes much more sense to first do the substitution and write $T$ explicitly as a function of $t$, then differentiate, rather than differentiate implicitly and then perform the substitution.
