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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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    Exactly. I've seen it.2011-08-12

3 Answers 3

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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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    oh, I see, sounds reasonable. thank you very much.2011-08-11
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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.2011-08-11
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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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    Oh yes. That is true.2011-08-12