Answer
$$dw = \frac{q}{{rs}}dp + \frac{p}{{rs}}dq - \frac{{pq}}{{s{r^2}}}dr - \frac{{pq}}{{{s^2}r}}ds$$
Work Step by Step
$$\eqalign{
& w = f\left( {p,q,r,s} \right) = \frac{{pq}}{{rs}} \cr
& {\text{Calculate the partial derivatives}} \cr
& {w_p} = \frac{q}{{rs}} \cr
& {w_q} = \frac{p}{{rs}} \cr
& {w_r} = \frac{{pq}}{s}\left( { - \frac{1}{{{r^2}}}} \right) = - \frac{{pq}}{{s{r^2}}} \cr
& {w_s} = \frac{{pq}}{r}\left( { - \frac{1}{{{s^2}}}} \right) = - \frac{{pq}}{{{s^2}r}} \cr
& {\text{The diferential }}dw{\text{ is given by}} \cr
& dw = {w_p}dp + {w_q}dq + {w_r}dr + {w_s}ds \cr
& dw = \frac{q}{{rs}}dp + \frac{p}{{rs}}dq - \frac{{pq}}{{s{r^2}}}dr - \frac{{pq}}{{{s^2}r}}ds \cr} $$
