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QUESTION

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

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ANSWER


$\ds f=x+y^2+\sin(xy)\\\frac{\partial f}{\partial
y}=2y+x\cos(xy)\\\frac{\partial^2f}{\partial x\partial
y}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial
y}\right)=0+\cos(xy)+x\left\{-\sin(xy)(y)\right\}=\cos(xy)-xy\sin(xy)$



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