Question

What is the derivative of `|x|`?

Answer 1

In general, there is no derivative as `f(x) = |x|` is not a continuous function.

However, some may find it useful to find the derivative of the function by breaking it into pieces and kind of "avoiding" the discontinuity.

`f'(x) = 1 \text( ; when x > 0)`
`f'(x) = -1 \text( ; when x < 0)`
`f'(x) \text( is undefined ; when x = 0)`

Explanation:

In general, there is no derivative as `f(x) = |x|` is not a continuous function.

However, some may find it useful to find the derivative of the function by breaking it into pieces and kind of "avoiding" the discontinuity.

If you consider the positive side of the function, it is simply a line of positive slope 1. So the derivative is 1.

If you consider the negative side of the function, it is simply a line of negative slope 1. So the derivative is -1.

Right at 0, the two one-side limits are contradictory and of course, this is the discontinuity so there is no derivative defined.

So you could summarize a piecewise derivative of `f(x) = |x|` as:
`f'(x) = 1 \text( ; when x > 0)`
`f'(x) = -1 \text( ; when x < 0)`
`f'(x) \text( is undefined ; when x = 0)`