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 $f_{yy}$?
Partial Derivatives: $f(x,y) = x^5y^4+\sin\left({x\over{y}}\right)$
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calculus
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0You meant partial _derivatives_ right? – 2012-11-26
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2Assuming, 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? – 2012-11-26
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0Yes, derivatives. I fixed it now. – 2012-11-26
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4Think about what $f_{xx}$ means. Let $F := f_x$. Then $f_{xx} = F_x$. Just take the partial derivative of the partial derivative. – 2012-11-26
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0I'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. – 2012-11-26
1 Answers
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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 $f_{yx}$