Real Analysis (derivative) I am completely stuck in this question.
If  $r &gt; 0$  is a rational number, let $f : \mathbb{R} \to \mathbb{R} $ be defined by $f (x)=x^r\sin (1/x)$ for $x \ne 0$, and $f (0)=0$. Determine those values of $r$ for which $f’(0)$
exists.
We look at the behaviour of the difference quotient as $h\to 0$ :

For $r&gt;1$ $$-h^{r-1}\leq \frac{h^r  \sin(1/h)-0}{h} \leq h^{r-1}$$ goes to $ 0$ as $h\to 0 $ .
So, $f'(0)=0$ .

For $ r= 1 $ , the difference quotient oscillates between $-1$ and $1$.

And lastly for $0&lt;r&lt;1$ ,the difference quotient oscillates unboundedly as $h\to 0$ .
hint:-  

$f'(x)=rx^{r-1}\sin \frac{1}{x}+x^{r-2} \cos\frac{1}{x}$   

now $f'(0)$ exists when $\lim _{x \to 0}x^{r-2} \cos\frac{1}{x}$ exist