## Calculus 8th Edition

$f_{xx}=12x^{2}y-12xy^{2}$ $f_{xy}=4x^{3}-12x^{2}y$ $f_{yx}=4x^{3}-12x^{2}y$ $f_{yy}=-4x^{3}$
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, $f_{xx}=12x^{2}y-12xy^{2}$ $f_{xy}=4x^{3}-12x^{2}y$ $f_{yx}=4x^{3}-12x^{2}y$ $f_{yy}=-4x^{3}$