Answer
$\frac{dF}{dT} = -\frac{8\pi^2~m~r}{T^3}$
Work Step by Step
We can find an expression for $v$ in terms of the period $T$:
$T = \frac{2\pi~r}{v}$
$v = \frac{2\pi~r}{T}$
We can find $\frac{dF}{dT}$:
$F = \frac{mv^2}{r}$
$F = \frac{m~(\frac{2\pi~r}{T})^2}{r}$
$F = \frac{4\pi^2~m~r}{T^2}$
$\frac{dF}{dT} = -\frac{8\pi^2~m~r}{T^3}$
