$\displaystyle f(t) = \frac{pqe^t}{(1-qe^t)^2}$
$\displaystyle \frac{d}{dt}f(t) = \frac{(pqte^t)(1-qe^t)^2 -2(1-qe^t)(-tqe^t)(pqe^t)}{(1-qe^t)^4}$ $\displaystyle = \frac{[(pqte^t)(1-qe^t)][(1-qe^t)+2(pqe^t)]}{(1-qe^t)^4}$
is this right?
$\displaystyle f(t) = \frac{pqe^t}{(1-qe^t)^2}$
$\displaystyle \frac{d}{dt}f(t) = \frac{(pqte^t)(1-qe^t)^2 -2(1-qe^t)(-tqe^t)(pqe^t)}{(1-qe^t)^4}$ $\displaystyle = \frac{[(pqte^t)(1-qe^t)][(1-qe^t)+2(pqe^t)]}{(1-qe^t)^4}$
is this right?