Answer
$ \dfrac{ds}{dt}=\dfrac{s-s^2e^{s^2t}}{2ste^{s^2t} -t}$
Work Step by Step
We have: $e^{s^2t}-st=1$
We differentiate both sides with respect to $t$.
$e^{s^2t} (s^2+2st \dfrac{ds}{dt})-s-t \dfrac{ds}{dt} =0 \\ s^2 e^{s^2t}+2st e^{s^2t} \dfrac{ds}{dt}-s-t \dfrac{ds}{dt}=0\\ (2ste^{s^2t} -t) \dfrac{ds}{dt}=s-s^2e^{s^2t}$
Therefore, $ \dfrac{ds}{dt}=\dfrac{s-s^2e^{s^2t}}{2ste^{s^2t} -t}$
