Answer
$$dz = 7dx - 2dy$$
Work Step by Step
$$\eqalign{
& z = 7x - 2y \cr
& {\text{Let }}z = f\left( {x,y} \right):{\text{ }} \cr
& {\text{Calculate the partial derivative }}{f_x}\left( {x,y} \right){\text{, treat }}y{\text{ as a constant}} \cr
& {\text{ }}{f_x}\left( {x,y} \right) = \frac{\partial }{{\partial x}}\left[ {7x - 2y} \right] \cr
& {\text{ }}{f_x}\left( {x,y} \right) = 7 \cr
& {\text{Calculate the partial derivative }}{f_y}\left( {x,y} \right){\text{, treat }}x{\text{ as a constant}} \cr
& {\text{ }}{f_y}\left( {x,y} \right) = \frac{\partial }{{\partial y}}\left[ {7x - 2y} \right] \cr
& {\text{ }}{f_y}\left( {x,y} \right) = - 2 \cr
& \cr
& {\text{The total differential of }}z{\text{ is given by }}dz = {f_x}\left( {x,y} \right)dx + {f_y}\left( {x,y} \right)dy \cr
& {\text{Substitute the partial derivatives }}{f_x}\left( {x,y} \right){\text{ and }}{f_y}\left( {x,y} \right) \cr
& dz = 7dx - 2dy \cr} $$
