$x z_x\+y z_y$

$\=x \left(y\+x f\prime w_x\+f\right)\+y \left(x\+x f\prime w_y\right)$

$\=x \left(y\+x f\prime \cdot \left(-y\/x^2\right)\+f\right)\+y \left(x\+x f\prime \cdot \left(1\/x\right)\right)$

$\=x y-y f\prime \+x f\+x y\+y f\prime$

$\=2 x y\+\left(y-y\right)f\prime \+x f$

$\=2 x y\+\+x f\+0\cdot f$

$\=2 x y_{}\+x f$

$\=2 x y\+\left(z-x y\right)$

$\=x y\+z$

Define $w$ and $z$ 

$w\=y\/x$$\stackrel{\text{assign}}{\to }$

$Z\=x y\+x f\left(w\right)$$\stackrel{\text{assign}}{\to }$

Compute $x z_x\+y z_y$ and simplify the result to $x y\+z$

$x \frac{\partial }{\partial x} Z\+y \frac{\partial }{\partial y} Z$ = $x\left(y\+f\left(\frac{y}{x}\right)-\frac{\mathrm{D}\left(f\right)\left(\frac{y}{x}\right)y}{x}\right)\+y\left(x\+\mathrm{D}\left(f\right)\left(\frac{y}{x}\right)\right)$$\stackrel{\text{expand}}{\=}$$2xy\+xf\left(\frac{y}{x}\right)$

Initialize

$w≔y\/x\:$

$Z≔x y\+x f\left(w\right)\:$

Apply the diff and simplify commands to evaluate the expression $x z_x\+y z_y$ 

$\mathrm{simplify}\left(x \mathrm{diff}\left(Z\,x\right)\+y \mathrm{diff}\left(Z\,y\right)\right)$ = $x\left(f\left(\frac{y}{x}\right)\+2y\right)$