I was asked to find when the function is differentiable and what is the derivative of:

$$\left\lfloor {{x^2}} \right\rfloor {\sin ^2}(\pi x)$$

Now, I am not sure how to treat the floor function.
I'll be glad for help.


marked as duplicate by Davide Giraudo, Thomas Andrews, TMM, Vedran Šego, rschwieb Jan 2 '14 at 13:51

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    $\begingroup$ The floor function is constant, except for jumps whenever $x^2$ is an integer. $\endgroup$ – user61527 Dec 31 '13 at 21:55

Hint: Split into two cases: Either $x^2$ is an integer, or it is not an integer. When $x^2\notin{\mathbb{Z}}$, $\lfloor x^2\rfloor$ will be a constant in some small interval around $x$, and so $$\frac{d}{dx} \lfloor x^2\rfloor\sin^2(\pi x)=\lfloor x^2\rfloor 2\pi \sin (\pi x) \cos(\pi x).$$

Try using the definition of the derivative to determine what happens when $x^2\in\mathbb{Z}$. There will be two cases depending on whether or not $x\in\mathbb{Z}$. In one of the cases, the derivative is not defined, and in the other case it will be zero.

  • $\begingroup$ So you're saying: $f(x) = \left\lfloor {{x^2}} \right\rfloor = f'(x)$. Right? $\endgroup$ – AndrePoole Dec 31 '13 at 22:12
  • $\begingroup$ @DanielFischer Right, corrected! Thanks! $\endgroup$ – Eric Naslund Dec 31 '13 at 23:04

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