Answer
$$
w_x= \frac{1}{y+z} ,\quad w_y= -x(y+z)^{-2},\quad w_z=-x(y+z)^{-2}.
$$
Work Step by Step
Recall that $(x^n)'=nx^{n-1}$
Since $ w=\frac{x}{y+z}=x(y+z)^{-1}$, we have
$$
w_x= \frac{1}{y+z} ,\quad w_y= -x(y+z)^{-2},\quad w_z=-x(y+z)^{-2}.
$$
