Chapter 3 - Applications of Differentiation - 3.1 Exercises - Page 167: 6

$f'(0)$ is undefined.

Using the rule that $\dfrac{d}{dx}|z(x)|=z'(x)\dfrac{|z(x)|}{z(x)}\to$ $f'(x)=0-(1\times\dfrac{|x|}{x})=-\dfrac{|x|}{x}; x\ne0.$ $f'(0)$ is undefined because the left-side limit of +1 does not agree with the right-hand limit of -1. This can also be seen from the graph.