Answer
$f_{x}(x, y, z) =\sin(y-z)$,
$f_{y}(x, y, z) =x\cos(y-z)$,
$f_{z}(x, y, z) =-x\cos(y-z)$
Work Step by Step
$f(x, y, z) =x\sin(y-z)$
Treat y and z as constant to calculate $f_{x}(x, y, z)$
$f_{x}(x, y, z) =\sin(y-z)(1)=\sin(y-z)$
Treat x and z as constant to calculate $f_{y}(x, y, z)$
$f_{y}(x, y, z) \stackrel{\text{chain rule} }{=}x\cos(y-z)(1)=x\cos(y-z)$
Treat x and y as constant to calculate $f_{z}(x, y, z)$
$f_{z}(x, y, z)\stackrel{\text{chain rule} }{=}x\cos(y-z) (-1)=-x\cos(y-z)$
