Small problem understanding application of chain rule

[K29]

Homework Statement

I have proven in two ways (correctly) that the derivative of ln|x| = 1/x (note absolute value does vanish)

Now I open my textbook and see a general rule that $$\frac{d}{dx} ln (u) = \frac{u'}{u}$$

And the not so general derivative of |x| is $$\frac{d}{dx} |x| = \frac{x}{|x|}$$

So using these statements $$\frac{d}{dx} ln (|x|) = \frac{|x|'}{|x|}$$
$$=\frac{(\frac{x}{|x|})}{|x|}$$
$$=\frac{x}{|x|^{2}}$$

I've looked over this a few times and I can't see what I've done wrong. I mean I'm looking for a silly mistake but I don't see it. Can you see where I've gone wrong? Whats going on here?

 

[Theaumasch]

abs(x)^2 = x^2