TeX source:
\begin{array}{rcl} f'_{xy}(x,y,z) &=& 150\cdot \left(-\dfrac{1}{2}\right)(x+yz)^{-\frac{1}{2}-1}\cdot z \\\\&=& -75z(x+yz)^{-\frac{3}{2}} \\\\&=& -\dfrac{75z}{\left(\sqrt{x+yz}\right)^3}\end{array}