# Partial Derivatives: $f(x,y) = x^5y^4+\sin\left({x\over{y}}\right)$

Given the function:$$f(x,y) = x^5y^4+\sin\left({x\over{y}}\right)$$ How can I find such partial derivatives as $f_{xx}$, $f_{xy}$, $f_{yx}$, and $f_{yy}$?

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You meant partial derivatives right? – EuYu Nov 26 '12 at 4:34
Assuming, as EuYu suggests, that you mean partial derivatives, you find them the same way you always find partial derivatives --- by differentiating with respect to one variable, while treating the other as a constant. Where are you getting stuck? – Gerry Myerson Nov 26 '12 at 4:36
Yes, derivatives. I fixed it now. – StealzHelium Nov 26 '12 at 4:38
Think about what $f_{xx}$ means. Let $F := f_x$. Then $f_{xx} = F_x$. Just take the partial derivative of the partial derivative. – anonymous Nov 26 '12 at 4:39
I'm getting stuck in what exactly the notation is asking me to do. But if all it's asking is to basically take it twice then I think I got it. – StealzHelium Nov 26 '12 at 4:41

You have $f(x,y) = x^5y^4+\sin\left({x\over{y}}\right)$:
• For $f_{xx}$ its asking you to take the derivative -- of the function -- twice of $x$ and treat $y$ as a constant. Same logic with $f_{yy}$
• For $f_{xy}$ its asking you to take the derivative -- of the function -- of $x$ treat $y$ as a constant, then take the derivative of $y$ and treat $x$ as a constant. Same logic $f_{yx}$