Calculus: Find the formula for f ' for the function f(x) = absolute value of (x - 6)?

Question

I've already proven f(x) is not differentiable at x by letting x assume values greater and less than 6. What I don't know how to find is the forumula for f '.

Answer 1

Remember that the absolute value function is just defined piecewise:

|x-6| = {x-6 if x≥6, -(x-6) if x<6}

So to find the derivative on the interior of any piece, just differentiate the formula on that piece:

d(|x-6|)/dx = {d(x-6)/dx if x>6, d(-(x-6))/dx if x<6}
d(|x-6|)/dx = {1 if x>6, -1 if x<6}

As you noted, d(|x-6|)/dx is not defined when x=6.

Answer 2

well, i dunno, but f'(x) = 1

x1 -6 = 1x0
=1