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