Question

Question #20d4f

Answer 1

`x/|x|`

Explanation:

To think of why this is the case, consider a couple different cases. First let's think about f(x) when x is less than 0.

Then, f(x) is a straight line with the slope -1. So the rate of change of f(x) at any point less than 0 is -1.

Now consider the case when x is greater than 0. It's similar, but with a slope of 1.

When x is zero, the derivative is undefined. This is because in the limit as x approaches zero from each side, the derivative has a different value (+1 from the positive side, -1 from the negative side). It's just like how the limit of a function is not defined if the function has two different limits when you approach it two different ways.

So
`f'(x) = -1` for `x<0, 1` for `x>0` and not defined at 0.

`f'(x)=x/|x|` fits all those requirements, and we could probably go into more detail about this but that's the idea. Make sense?