Answer
$f$ is not differentiable at $x = 6$
$f'(x) = -1~~~~$ if $x \lt 6$
$f'(x) = 1~~~~~$ if $x \gt 6$
We can see a sketch of $f'(x)$ below.
Work Step by Step
$f(x) = \vert x-6 \vert$
$f'(a) = \lim\limits_{x \to a}\frac{\vert x-6 \vert-\vert a-6 \vert}{x-a}$
$\lim\limits_{x \to 6^-}\frac{\vert x-6 \vert-\vert 6-6 \vert}{x-6}$
$=\lim\limits_{x \to 6^-}\frac{-(x-6)-0}{x-6}$
$= -1$
$\lim\limits_{x \to 6^+}\frac{\vert x-6 \vert-\vert 6-6 \vert}{x-6}$
$=\lim\limits_{x \to 6^+}\frac{(x-6)-0}{x-6}$
$= 1$
Since the left limit does not equal the right limit at $x = 6$, $f'(6)$ does not exist.
Then $f$ is not differentiable at $x = 6$
We can find an expression for $f'(a)$ when $a \lt 6$:
$\lim\limits_{x \to a}\frac{\vert x-6 \vert-\vert a-6 \vert}{x-a}$
$=\lim\limits_{x \to a}\frac{-(x-6)-(6-a)}{x-a}$
$=\lim\limits_{x \to a}\frac{a-x}{x-a}$
$= -1$
We can find an expression for $f'(a)$ when $a \gt 6$:
$\lim\limits_{x \to a}\frac{\vert x-6 \vert-\vert a-6 \vert}{x-a}$
$=\lim\limits_{x \to a}\frac{(x-6)-(a-6)}{x-a}$
$=\lim\limits_{x \to a}\frac{x-a}{x-a}$
$= 1$
Therefore:
$f'(x) = -1~~~~$ if $x \lt 6$
$f'(x) = 1~~~~~$ if $x \gt 6$
We can see a sketch of $f'(x)$ below.
