Answer
$f_{x}(x, y, z) =z-10xy^{3}z^{4},$
$f_{y}(x, y, z) =-15x^{2}y^{2}z^{4}$,
$f_{z}(x, y, z)=x-20x^{2}y^{3}z^{3}$
Work Step by Step
$f(x, y, z) =xz -5x^{2}y^{3}z^{4}$
Treat y and z as constant to calculate $f_{x}(x, y, z)$
$f_{x}(x, y, z) =z-5y^{3}z^{4}(2x)=z-10xy^{3}z^{4}$
Treat x and z as constant to calculate $f_{y}(x, y, z)$
$f_{y}(x, y, z) =0-5x^{2}z^{4}(3y^{2})=-15x^{2}y^{2}z^{4}$,
Treat x and y as constant to calculate $f_{z}(x, y, z)$
$f_{z}(x, y, z)=x-5x^{2}y^{3}(4z^{3})=x-20x^{2}y^{3}z^{3}$
