Answer
$\frac{x-6}{\left|x-6\right|}$
Work Step by Step
Step 1:
Use the chain rule $\frac{df\left(a\right)}{dx}=\frac{df}{da}\cdot \frac{da}{dx}$
$f=\left|a\right|,\:\:a=x-6$ $=\frac{d}{da}\left(\left|a\right|\right)\frac{d}{dx}\left(x-6\right)$
$\frac{d}{da}\left(\left|a\right|\right)=\frac{a}{\left|a\right|}$
Step 2:
Use the Sum Difference rule: $\frac{d}{dx}\left(x-6\right)$$=\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(6\right)$
$\frac{d}{dx}\left(x\right)=1$
Step 3:
Since the derivative of a constant is zero than: $\frac{d}{dx}\left(6\right)=0$
$=\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(6\right)$ = 1 - 0 = 1
Remember we used $a = x-6$ so substitute in for $a$
$\frac{a}{\left|a\right|}\cdot \:1$ = $\frac{x-6}{\left|x-6\right|}\cdot \:1$
Since anything multiplied by 1 is itself, we can simply forget about the 1 and simplify $\frac{x-6}{\left|x-6\right|}\cdot \:1$ to $\frac{x-6}{\left|x-6\right|}$
Answer: $\frac{x-6}{\left|x-6\right|}$
