Question

How do you find f'(x) using the definition of a derivative for `f(x)=absx`?

Answer 1

See the explanation.

Explanation:

The function `f(x)=|x|` is continuous function, defined for `AAx in R` but it's not differentiable for `AAx in R` (e.g. derivative doesn't exist in every point).

By definition, function is differentiable at a point if its derivative exists at that point.

`f(x)=x` for `x>=0`
`f(x)=-x` for `x<0`

For `x>0`:
`f'(x)=lim_(h->0) (x+h-x)/h = lim_(h->0) h/h =lim_(h->0) 1 = 1`

For `x<0`:
`f'(x)=lim_(h->0) (-(x+h)-(-x))/h = lim_(h->0) (-x-h+x)/h`

`f'(x) =lim_(h->0) (-h)/h = lim_(h->0) (-1) = -1`

From the previous:

`f'(0+epsilon) = 1`
`f'(0-epsilon) = -1`

`f'(0+epsilon) != f'(0-epsilon)`

Function is not differentiable at `x=0`.