Question

This function Is it differentiable at x = 1​ ?

Answer 1

It is not differentiable at x=1 since the slope of the tangent line as x -> 1 from the right is 1 while the slope of the tangent line as x->1 from the left is -1

Answer 2

By definition of absolute value:

$|x - 1| = \begin{cases} x - 1 & \text{if } x \ge 1 \\ -(x-1) & \text{if } x < 1 \end{cases}$

Then the derivative is

$\dfrac{d|x-1|}{dx} = \begin{cases} 1 & \text{if } x > 1 \\ -1 & \text{if } x < 1\end{cases}$

but does not exist at $x=1$ because

$\displaystyle \lim_{x\to1^-} f'(x) = -1$

while

$\displaystyle \lim_{x\to1^+} f'(x) = 1$

and these limits are not equal, so the derivative is discontinuous and hence $|x-1|$ is not differentiable at $x=1$.