# Domain of Ln(Ln(sinx)) and its derivative.

Im reading the Ian Stewart book Concepts of Modern Mathematics, in a page he talks about the function $\ f(x) = log(log(sin(x)))$ and its derivate, $\ f'(x) = \frac{ cot(x)}{log(sin(x))}$. Now, you can´t sketch the graphic of $\ f$ because for all values of $x$, $f(x)$ doesn´t exist. However, $f'$ does make sense for all $x$ such that $sin(x)>0$. If the derivative of a function is the slope of the tangent of every point, but the function $f$ hasn´t points, where is the mistake? if the domain of any function $g$ is the empty set, the function automatically hasn´t derivative?

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This plot may help show how it has an imaginary and real part for all values of $x$: wolframalpha.com/input/… – apnorton Jan 8 '13 at 3:41
ok, i understand the imaginary part, of course, but, why the plot has a "real part"? for all values of x, i can´t calcule f(x), why Wolfram shows me a graphic of f in blue? – dwarandae Jan 8 '13 at 3:46
It's still complex for the whole graph. Complex numbers are written in the form $a + bi$. $a$ is the real part, and $b$ is the imaginary part. When the plot says it has a real part, that means $a$ is non-zero. However, $f(x)$ is still complex (that is, $b$ is also non-zero). – apnorton Jan 8 '13 at 3:59

There's no mistake: the function $\,f'(x)\,$ exists on its own, without any regard of the fantasy called $\,f(x)\,$ which, as you note, doesn't exist...as a real function.
In fact, the complex function $\,f(x)\,\,,\,x\in\Bbb C\,$ exists pretty nicely, but since it seems like you haven't yet studied complex analysis I'm stopping here.
$$f(x) = \log | \log( \sin(x))|$$ works for $0 < x < \frac{\pi}{2}$