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I have a differential of a function, $df(x,y)=y\sin(xy)\mathrm{dx}+x\sin(xy)\mathrm{dy}$.

How do I determine the double partial derivatives $f_{xy}, f_{xx}$ and $f_{yy}$?

I am fairly certain I have to use the chain rule, but I can't see how to apply because of the $y$ and $x$ in the front.

If it's true that I should use the chain rule, could you give me a hint?

1 Answers 1

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Given
$df(x,y)=y\sin(xy)\mathrm{dx}+x\sin(xy)\mathrm{dy}$

By Using
$df(x,y)=f_{x}\mathrm{dx}+f_{y}\mathrm{dy}$

We can get that
$f_{x}=y\sin(xy)$
$f_{y}=x\sin(xy)$

From there:
$f_{xx}=y^{2}\cos(xy)$,
$f_{yy}=x^{2}\cos(xy)$
$f_{xy}=\sin(xy)+xy\cos(xy)$.

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    This answer seems to me to be "too much of an answer" for a homework question. I thought we were only supposed to give hints?2012-10-09