TeX source:
\begin{array}{rcl} f'_{zy}(x,y,z) &=& \dfrac{2xy e^{-3xy}+xy^2 e^{-3xy} \cdot(-3x)}{z} \\\\&=& \dfrac{2xy e^{-3xy}-3x^2y^2 e^{-3xy}}{z} \\\\&=& \dfrac{2xy-3x^2y^2}{z} e^{-3xy} \\\\\end{array}