# Find all the second partial derivatives. f(x,y)=x^{4}y-2x^{5}y^{2} f_{xx}(x,y)= f_{xy}(x,y)= f_{yx}(x,y)= f_{yy}(x,y)=

Find all the second partial derivatives.
$f\left(x,y\right)={x}^{4}y-2{x}^{5}{y}^{2}$
${f}_{×}\left(x,y\right)=$
${f}_{xy}\left(x,y\right)=$
${f}_{yx}\left(x,y\right)=$
${f}_{yy}\left(x,y\right)=$
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Step 1
Given function is $f\left(x,y\right)={x}^{4}y-2{x}^{5}{y}^{2}$
Obtain the first partial derivative with respect to x and y.
${f}_{x}\left(x,y\right)={\left({x}^{4}y-2{x}^{5}{y}^{2}\right)}^{\prime }$
$=4{x}^{3}y-10{x}^{4}{y}^{2}$
${f}_{y}\left(x,y\right)={\left({x}^{4}y-2{x}^{5}{y}^{2}\right)}^{\prime }$
$={x}^{4}-4{x}^{5}y$
Step 2
Compute the second derivative.
${f}_{×}\left(x,y\right)={\left(4{x}^{3}y-10{x}^{4}{y}^{2}\right)}^{\prime }$
$=12{x}^{2}y-40{x}^{3}{y}^{2}$
${f}_{xy}\left(x,y\right)={\left(4{x}^{3}y-10{x}^{4}{y}^{2}\right)}^{\prime }$
$=4{x}^{3}-20{x}^{4}y$
${f}_{yx}\left(x,y\right)={\left({x}^{4}-4{x}^{5}y\right)}^{\prime }$
$=4{x}^{3}-20{x}^{4}y$
${f}_{yy}={\left({x}^{4}-4{x}^{5}y\right)}^{\prime }$
$=0-4{x}^{5}$
$=-4{x}^{5}$