Answer
$$dw = \frac{1}{{y + z}}du + \frac{1}{{y + z}}dx - \frac{{u + x}}{{{{\left( {y + z} \right)}^2}}}dy - \frac{{u + x}}{{{{\left( {y + z} \right)}^2}}}dz$$
Work Step by Step
$$\eqalign{
& w = f\left( {u,x,y,z} \right) = \frac{{u + x}}{{y + z}} \cr
& {\text{Calculate the partial derivatives}} \cr
& {w_u} = \frac{1}{{y + z}} \cr
& {w_x} = \frac{1}{{y + z}} \cr
& {w_y} = - \frac{{u + x}}{{{{\left( {y + z} \right)}^2}}} \cr
& {w_z} = - \frac{{u + x}}{{{{\left( {y + z} \right)}^2}}} \cr
& {\text{The diferential }}dw{\text{ is given by}} \cr
& dw = {w_u}du + {w_x}dx + {w_y}dy + {w_y}dz \cr
& dw = \frac{1}{{y + z}}du + \frac{1}{{y + z}}dx - \frac{{u + x}}{{{{\left( {y + z} \right)}^2}}}dy - \frac{{u + x}}{{{{\left( {y + z} \right)}^2}}}dz \cr} $$
