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So. I have to show that this functiun is differentiable outside of $(0,0)$ and also to calculate its differentiation. $$f(x,y)=xy\frac{x^2-y^2}{x^2+y^2}$$ when $(x,y)$ is not equal to $(0,0)$ and $$f(x,y)=0$$ when $(x,y)$ is equal to $(0,0)$. Also we have to show that the function is differentiable on $(0,0)$ too. $$$$ Firstly I wanted to see if the functiun is continuous e.g: $$\lim_{(x,y)\to (0,0)}{xy\frac{x^2-y^2}{x^2+y^2}}=0$$ so it is countinous.$$$$ Than I wanted to see if it is differentiable but I don't know how to continue, all I got is: $$\lim_{(x,y)\to (0,0)} \frac{f(x,y)-f(0,0)}{(x,y)-(0,0)}=\lim_{(x,y)\to (0,0)}\frac{xy\frac{x^2-y^2}{x^2+y^2}}{\sqrt{x^2+y^2}}$$ Can someone put me on the right track? And explain me what am I doing wrong? I don't know how to continue it..

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    For any $\;(x,y)\neq(0,0)\;$ , the function is a rational one for which the denominator doesn't vanish and is thus everywhere continuous, differentiable and etc. – 2017-02-01
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    @DonAntonio can you develop? I mean, if the denominator does not vanish is differentiable? And what about when $(x,y)=(0,0)$ I mentioned that too.. I have to check if it is differentiable in $(0,0)$ too.. – 2017-02-01
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    Similar to (http://math.stackexchange.com/q/1659129). – 2017-02-01

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Hint. By changing to polar coordinates one gets, as $(x,y) \to (0,0)$, $$ \left|\frac{f(x,y)-f(0,0)}{(x,y)-(0,0)}\right|=\left|\frac{xy\frac{x^2-y^2}{x^2+y^2}}{\sqrt{x^2+y^2}}\right|\le\rho\cdot|\cos \theta \cdot \sin \theta| \cdot|\cos^2 \theta-\sin^2 \theta|\le 2\rho \to 0 $$ yielding the desired differentiability.

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    I still don't get it.. Can you develop? What properties a function should have to be differentiable(You lost me in this rush explenation)? – 2017-02-01
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    As $(x,y) \to (0,0)$, you just want the quotient on the left hand side to become smaller and smaller... which is the case since $\rho=\sqrt{x^2+y^2}$. – 2017-02-01
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    Ok.. I still want to know what properties a function should have to be differentiable? I mean i dont get it why you had to compare that module with $\rho^{3/2}|\cos \theta \cdot \sin \theta| \cdot|\cos^2 \theta-\sin^2 \theta$ – 2017-02-01
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Observe

$$\tag 1\left |\frac{xy\frac{x^2-y^2}{x^2+y^2}}{\sqrt{x^2+y^2}}\right| \le \frac{|xy|\cdot 1}{\sqrt{x^2+y^2}}.$$

Since $|xy| \le (x^2+y^2)/2,$ the limit in $(1)$ is $0.$