Answer
$f_{xx}=12x^{2}y-12xy^{2}$
$f_{xy}=4x^{3}-12x^{2}y$
$f_{yx}=4x^{3}-12x^{2}y$
$f_{yy}=-4x^{3}$
Work Step by Step
Consider the function $f(x,y)=x^{4}y-2x^{3}y^{2}$
Differentiate the function with respect to $x$ keeping $y$ constant.
$f_{x}=4x^{3}y-6x^{2}y^{2}$
Differentiate $f_{x}$ with respect to $x$ keeping $y$ constant.
$f_{xx}=12x^{2}y-12xy^{2}$
Differentiate $f_{x}$ with respect to $y$ keeping $x$ constant.
$f_{xy}=4x^{3}-12x^{2}y$
Differentiate the function with respect to $y$ keeping $x$ constant.
$f_{y}=x^{4}-4x^{3}y$
Differentiate $f_{y}$ with respect to $x$ keeping $y$ constant.
$f_{yx}=4x^{3}-12x^{2}y$
Differentiate $f_{y}$ with respect to $y$ keeping $x$ constant.
$f_{yy}=-4x^{3}$
Hence, $f_{xx}=12x^{2}y-12xy^{2}$
$f_{xy}=4x^{3}-12x^{2}y$
$f_{yx}=4x^{3}-12x^{2}y$
$f_{yy}=-4x^{3}$
