Answer
$g_x=\frac{2x^2}{z^2+x^2}+ln(z^2+x^2)$
$g_z=\frac{2xz}{z^2+x^2}$
Work Step by Step
Take the first partial derivatives of the given function. When taking partial derivative with respect to x, treat z as a constant, and vice versa:
$g_x=x\times\frac{2x}{z^2+x^2}+ln(z^2+x^2)=\frac{2x^2}{z^2+x^2}+ln(z^2+x^2)$
$g_z=x\times\frac{2z}{z^2+x^2}=\frac{2xz}{z^2+x^2}$
