Can somebody provide an explanation to the formula of a one elementary integral?

Question

Here is the formula:

$$ \int{\frac{dx}{x}} = \ln{|x|} + C $$

In my textbook it is given without proof, so I have a little confusion here. From the definition of integral this equality must be true:

$$ (\ln|x| + C)` = \frac{1}{x} $$

But I failed to derive it. Cause according to the rule of differentiation of a complex funtion $(f_1(f_2(x)))` = f_1`(f_2(x))*f_2`(x) $ (if I understand it right) the derivative of $\ln|x| + C$ is:

$$ (\ln|x| + C)` = (\ln|x|)` + C` = \frac{1}{|x|} * |x|` + 0 $$

And this is confirmed by the Wolfram Mathematica:

So this is obviously not $\frac{1}{x}$. Can somebody provide an explanation for this problem?

My appreciation.

Answer 1

$|x|'=Sgn(x)$ and $\frac{\operatorname{Sgn}(x)}{|x|}=\frac{1}{x}$, where $\operatorname{Sgn(x)}=1$ if $x\geq 0$, and $\operatorname{Sgn(x)}=-1$ if $x< 0$