<s>[INST] A question from Munkres' Analysis on Manidolds (P. 55, E. 10) 
*

*Define $f: \mathbf{R}^{2} \rightarrow \mathbf{R}$ by setting $f(\mathbf{0})=0$, and
$$
f(x, y)=x y\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right) \text { if }(x, y) \neq 0
$$
i) Show $f$ is of class $C^{1}$ on $\mathbf{R}^{2} .$ [Hint: Show $D_{1} f(x, y)$ equals the product of $y$ and a bounded function, and $D_{2} f(x, y)$ equals the product
of $x \text { and a bounded function. }]$
My Attempt. $D_1 f(x,y)=\frac{d}{d x}\left(\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}\right)=\frac{y\left(x^{4}+4 x^{2} y^{2}-y^{4}\right)}{\left(x^{2}+y^{2}\right)^{2}}$ and $D_2 f(x,y)=\frac{d}{d y}\left(\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}\right)=\frac{x\left(-y^{4}-4 x^{2} y^{2}+x^{4}\right)}{\left(x^{2}+y^{2}\right)^{2}}.$
Now, how can I use the hint, may you help? Thanks...
Can we say $\frac{\left(x^{4}+4 x^{2} y^{2}-y^{4}\right)}{\left(x^{2}+y^{2}\right)^{2}}$ is a bounded function in $D_1 f(x,y)$?
 [/INST] Switch to polar coordinates: 
$D_1 f(x,y)=\frac{d}{d x}\left(\frac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}\right)=\frac{y\left(x^{4}+4 x^{2} y^{2}-y^{4}\right)}{\left(x^{2}+y^{2}\right)^{2}}=y\left(\frac{r^4(\cos^4 t+4\cos^2 t \sin^2 t-\sin^4t )}{r^4}\right)\le 6y$. 
Similarly for $D_2f(x,y)$
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