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I want to solve $\displaystyle f_1(x,y)$ where $f(x,y)= \dfrac{x}{x^2+y^2}$ by hand.

What steps are involved?

Update: $\displaystyle f_1(x,y)$ denotes $\dfrac{\partial}{\partial x}f(x,y)$

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with respect to $x$, $y$ or some other variable? – enzotib Sep 27 '12 at 16:31
Updated question. – Farmor Sep 27 '12 at 16:41
What exactly do you want? Is it to find a function $f(x,y)$ such that ${\partial\over\partial x} f(x,y)={x\over x^2+y^2}$? – David Mitra Sep 27 '12 at 16:43
@DavidMitra Why the down vote? I want do derive with regards to x and have successfully done so with the help of the quotient rule. – Farmor Sep 27 '12 at 16:47
I didn't downvote. But I suspect whoever did it, did so because your question is not all all clear. If you wish to compute a partial derivative, you should ask: "Let $f(x,y)={x\over x^2+y^2}$. How do I find $f_1(x,y)$, where $f_1$ denotes ..." Do not assume people know what your notation means. The way your question is currently phrased led me to my previous comment. – David Mitra Sep 27 '12 at 16:51
up vote 2 down vote accepted

You use the quotient rule to get $\left(\frac x{x^2+y^2}\right)'=\frac {x'(x^2+y^2)-(x^2+y^2)'x}{(x^2+y^2)^2}$ then take the indicated derivatives in the numerator.

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Thanks I solved it beautifully with the quotient rule. – Farmor Sep 27 '12 at 16:58

Treat $y$ like a constant and use quotient rule to derive with respect to $x$. Same idea for deriving with respect to $y$.

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