Question

For what values of x is the function `f(x)=abs(x^2-9)` differentiable?

Answer 1

`f(x)=|x^2-9|` is differentiable everywhere with the exception of `x=+-3`

Explanation:

Although `x^2-9` is both continuous and differentiable everywhere the same is not true for `|x^2-9|`, which is continuous everywhere but not differentiable at the transition between positive and negative.

This transition occurs when:

`|x^2-9| = 0 => x^2-9=0`
`" "=> x^2=9`
`" "=> x = +- 3`

Hence the function `f(x)=|x^2-9|` is differentiable everywhere with the exception of `x=+-3`

graph{|x^2-9| [-20.04, 19.96, -3.16, 16.84]}