Answer
$f_{x}(x, y)=y^{2}-3x^{2}y$
$f_{y}(x, y)=2xy-x^{3}$
Work Step by Step
$f(x, y) =xy^{2}-x^{3}y$
$f_{x}(x, y) =\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h,y)-f(x,y)}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{(x+h)y^{2}-(x+h)^{3}y-(xy^{2}-x^{3}y)}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{xy^{2}+hy^{2}-(x^{3}y+3x^{2}hy+3xh^{2}y+h^{3}y)-xy^{2}+x^{3}y}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{xy^{2}+hy^{2}-x^{3}y-x^{2}hy-3xh^{2}y-h^{3}y-xy^{2}+x^{3}y}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{h(y^{2}-3x^{2}y-3xyh-yh^{2})}{h}$
$=\displaystyle \lim_{h\rightarrow 0}(y^{2}-3x^{2}y-3xyh-yh^{2})$
$=y^{2}-3x^{2}y$
$f_{y}(x, y)=\displaystyle \lim_{h\rightarrow 0}\frac{f(x,y+h)-f(x,y)}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{x(y+h)^{2}-x^{3}(y+h)-(xy^{2}-x^{3}y)}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{xy^{2}+2xyh+xh^{2}-x^{3}y-x^{3}h-xy^{2}+x^{3}y}{h}$
$=\displaystyle \lim_{h\rightarrow 0}\frac{h(2xy+xh-x^{3})}{h}$
$=\displaystyle \lim_{h\rightarrow 0}(2xy+xh-x^{3})$
$=2xy-x^{3}$
