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Let $$u(x,y)=\frac{x}{x^2+y^2}$$

I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial derivative of $y$, equal $0$.

However, I'm kind of stuck. I'm using the quotient rule to solve for the partial derivative of $x$, but is this the right way to take a partial derivative of a quotient?

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    $\begingroup$ This is not a harmonic analysis question. Please read tag descriptions before using. $\endgroup$ Commented Feb 2, 2016 at 2:11
  • $\begingroup$ @MattRosenzweig this is not possible on the mobile app, my apologies $\endgroup$ Commented Feb 2, 2016 at 2:13

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When you take a partial derivative of a multivariate function, you are simply "fixing" the variables you don't need and differentiating with respect to the variable you do. Thus since you have a rational function with respect to $x$, you simply fix $y$ and differentiate using the quotient rule.

Now, you can use the product rule if you choose; you just have to rewrite:

$$u(x,y)=\frac{x}{x^2+y^2}=x\left(x^2+y^2\right)^{-1}$$

Then $$\frac{\partial{u}}{\partial{x}}=x\cdot(-2x)(x^2+y^2)^{-2}+(x^2+y^2)^{-1}$$

which you can simplify.

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