Answer
$f_{xyx}=24x^{2}y-6x$
$f_{xxx}=24xy^{2}-6y$ and $f_{xyx}=24x^{2}y-6x$
Work Step by Step
Consider the function $f(x,y)=x^{4}y^{2}-x^{3}y$
Let us start by finding $f_{x}(x,y)$ by differentiating $f(x,y) $with respect to $x$ keeping $y$ constant.
As we know
$f_{x}=\frac{∂}{∂x}f(x,y) $
$=\frac{∂}{∂x}[x^{4}y^{2}-x^{3}y]$
$=4x^{3}y^{2}-3x^{2}y$
$f_{xx}=\frac{∂}{∂x}[4x^{3}y^{2}-3x^{2}y]=12x^{2}y^{2}-6xy$
Thus, $f_{xxx}=\frac{∂}{∂x}[12x^{2}y^{2}-6xy]=24xy^{2}-6y$
$f_{xy}=\frac{∂}{∂y}[4x^{3}y^{2}-3x^{2}y]=8x^{3}y-3x^{2}$
$f_{xyx}=\frac{∂}{∂x}[8x^{3}y-3x^{2}]$
Hence, $f_{xyx}=24x^{2}y-6x$
$f_{xxx}=24xy^{2}-6y$ and $f_{xyx}=24x^{2}y-6x$
