Partial derivative deduction.

Question

Let $f(x,y)=xy\frac{x^2-y^2}{x^2+y^2}$

Calculate $\frac{\partial{f}}{\partial x}$ and deduce $\frac{\partial{f}}{\partial y}$

I've done the calculations for $\frac{\partial{f}}{\partial x}$ and got $\frac{\partial{f}}{\partial x}$=$\frac{x^4y+4x^2y^3-y^5}{(x^2+y^2)^2}$

However i have no idea how to deduce $\frac{\partial{f}}{\partial y}$.

I would be grateful if someone could help me.

Thanks in advance.

Answer 1

hint...note that $f(x,y)=-f(y,x)$ so you should just get the negative of the same expression but with the $x$ and the $y$ interchanged.

Answer 2

You will get $$\frac{\partial f(x,y)}{\partial x}=\frac{y \left(x^4+4 x^2 y^2-y^4\right)}{\left(x^2+y^2\right)^2}$$ and $$\frac{\partial f(x,y)}{\partial y}=\frac{x \left(x^4-4 x^2 y^2-y^4\right)}{\left(x^2+y^2\right)^2}$$