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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$\begingroup$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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