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\newcommand\ds{\displaystyle}
\begin{document}

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QUESTION

Given $f(x,y)=x^2y^2+e^{xy}$ evaluate
$\ds\frac{\partial^2f}{\partial y\partial x}$.



ANSWER

$\ds f=x^2y^2+e^{xy},\ \frac{\partial f}{\partial
x}=2xy^2+ye^{xy}\\ \frac{\partial^2f}{\partial y\partial
x}=\frac{\partial}{\partial y}(2xy^2+ye^{xy})=4xy+e^{xy}+xye^{xy}$




\end{document}