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How I can calculate the derivative of $$f(x) = \left\{ \begin{gathered} {x^2}\quad,\quad{\text{if}}\quad x \in \mathbb{Q} \\ {x^3}\quad,\quad{\text{if}}\quad x \notin \mathbb{Q} \\ \end{gathered} \right.$$ at some $x\in \mathbb{R}$?

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There are actually two points where this function is continuous: the two points where the graphs of the cubic and quadratic functions intersect. But it's differentiable at only one of them. –  Michael Hardy Aug 11 '11 at 19:41
    
Exactly. I've seen it. –  mathsalomon Aug 12 '11 at 15:41
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3 Answers 3

HINT:

The derivative exists if $\lim _{y \to x} \dfrac{f(y) - f(x)}{y - x}$ exists. Of course, a limit must be the same along any Cauchy sequence. So at what points does the derivative even exist? (it does exist somewhere)

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Thanks for the reply, but how can calculate such a limit?. If you only know where the variable estimate limits on the numbers approaching real. –  mathsalomon Aug 11 '11 at 6:25
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@mathsalomon For example, suppose we wanted to calculate it at $x = 2$. Then on irrationals, the limit is 4. On the irrationals, the limit is 12. They're not equal, so it's not differentiable at 2. –  mixedmath Aug 11 '11 at 6:27
    
oh, I see, sounds reasonable. thank you very much. –  mathsalomon Aug 11 '11 at 6:37
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The first helpful information to look for is if your function is continuous at any $x$. After all, a function does not have a well-defined derivative where it isn't continuous.

Then, analyze those points where it is continuous. Does it have a derivative there? A hint is that there is always a rational point in between two real numbers (that aren't equal) and that there's always an irrational point in between two real numbers (again, nonequal).

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I think your very good suggestion, I think I will do so. thanks. –  mathsalomon Aug 11 '11 at 6:29
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What makes you think it has a derivative? Doesn't a function have to be continuous to be differentiable?

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If you plot the function, we could be continued only at $x=0$ and $x=1$. –  mathsalomon Aug 11 '11 at 6:33
    
If what you are saying is that the function is continuous only at $x=0$ and $x=1$, then, yes, that's true. –  Gerry Myerson Aug 11 '11 at 13:11
    
Oh yes. That is true. –  mathsalomon Aug 12 '11 at 15:44
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