A differentiation step in a economic growth model
You have most of this; it seems like you might have a miscalculation when differentiating the fraction:
$$
\begin{align*}
\frac{dk_t}{dt} = \frac{d\frac{K_t}{N_t}}{dt} &amp; = \frac{\frac{dK_t}{dt}N_t - \frac{dN_t}{dt}K_t}{N_t^2} \\
&amp; = \frac{\frac{dK_t}{dt}}{N_t} - nk_t
\end{align*}
$$
thus
$$
\frac{\frac{dK_t}{dt}}{N_t} = \frac{dk_t}{dt} + nk_t,
$$
and you can plug this into your equation (4).