What is the derivative of the next equation?

$$f(x) = \left\lfloor\frac{c}{x}\right\rfloor\ \ \ \ \ \text{ where }\lfloor\cdot\rfloor\text{ is the floor function.}$$

  • $c,x$ are positive integers
  • floor is a function that convert float number to integer by removing everything after the dot. Sorry I do no know how else to explain it.

Does floor plays here any roll or it will be equal to derivative and then flooring the result.


closed as off-topic by dustin, Will Jagy, MPW, Algebraic Pavel, Claude Leibovici Jan 24 '15 at 4:43

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  • 2
    $\begingroup$ Note that you have specified that both $c$ and $x$ are integers. In this case, the domain is discrete. Do you think the derivative is meaningful in such cases? $\endgroup$ – MPW Jan 24 '15 at 2:50
  • $\begingroup$ @MPW sorry, what is domain and what is discrete? $\endgroup$ – Ilya Gazman Jan 24 '15 at 4:05
  • $\begingroup$ Domain of $f$ is the set of values for which $f$ is defined. Discrete means it is a set of isolated points with gaps between. Derivative only makes sense on dense set. $\endgroup$ – MJD Jan 29 '15 at 15:00
  • $\begingroup$ The derivative of floor(x) is universally equal to 0*(x%1/x%1). Unfortunately you cannot cancel as there are places where division by 0 occurs. If you ignore that discontinuity then derivative of any floor function is 0, which is about as useful as saying floor is a constant. $\endgroup$ – The Great Duck Mar 4 '16 at 5:57

$f'(x) = \begin{cases} \mathrm {not \quad differentiable , if\quad} x=\frac cn \quad \mathrm for\ some\ positive \ integer\ n \\ 0 , x = other \ points\end{cases}$

Hint: Just do it by the definition of derivative.( It is not difficult ,so I omit the proof.)

  • $\begingroup$ The question is about the floor effect. Can you explain why do you think it will have no effect? $\endgroup$ – Ilya Gazman Jan 24 '15 at 2:40
  • 2
    $\begingroup$ @Ilya Have you plotted this function? It should be fairly clear that it is composed of flat bits (with derivative $0$) and discontinuities (with no derivative) from a graph. $\endgroup$ – Milo Brandt Jan 24 '15 at 2:49
  • $\begingroup$ @Ilya_Gazman Sorry,I don't get your point. what you means "have no effect".$f(x)=\frac cx$ is defferentiable everywhere on $(0,\infty )$ but floor function not. $\endgroup$ – Syuizen Jan 24 '15 at 2:49
  • $\begingroup$ why floor is not deferential? And in the areas that it is deferential, will it provide the same result as if I apply it after the deferential? $\endgroup$ – Ilya Gazman Jan 24 '15 at 2:52

As Syuizen points out, $\lfloor x \rfloor$ is not differentiable as a function.

However, as a distribution, you can write the derivative of $\lfloor x \rfloor$ as an infinite sum of Dirac delta functions:

$$\frac{d}{dx} \lfloor x \rfloor = \sum_{n \in \mathbb Z} \delta_n(x).$$


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