Differentiate the modulus function.

Question

How do I differentiate $ f(x)= 1+|x-2| $ ? How to deal with modulus sign here?

Answer 1

If $x>1$, $f(x)=1+(x-2)=x-1$ and $f'(x)=1$

If $x<1$, $f(x)=1-(x-2)=3-x$ and $f'(x)=-1$.

$\displaystyle \lim_{h\to0^+}\frac{f(2+h)-f(2)}{h}=\lim_{h\to0^+}\frac{[(2+h)-1]-1}{h}=1$ and $\displaystyle \lim_{h\to0^-}\frac{f(2+h)-f(2)}{h}=\lim_{h\to0^-}\frac{[3-(2+h)]-1}{h}=-1$.

Therefore $\displaystyle \lim_{h\to0}\frac{f(2+h)-f(2)}{h}$ does not exist. $f$ is not differentiable at $x=2$.