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 $x\in \mathbb{R}$?
How I can calculate the derivative of a piecewise function like this?
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0Exactly. I've seen it. – 2011-08-12
3 Answers
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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0oh, I see, sounds reasonable. thank you very much. – 2011-08-11
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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0I think your very good suggestion, I think I will do so. thanks. – 2011-08-11
What makes you think it has a derivative? Doesn't a function have to be continuous to be differentiable?
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0Oh yes. That is true. – 2011-08-12