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