# Derivative of |x|

1. Jan 19, 2010

### Char. Limit

Is this: $$\frac{d |x|}{dx}$$ equal to the unit step function?

Also, is the unit step function equal to $$y=\frac{|x|}{x}=\frac{x}{|x|}$$?

2. Jan 19, 2010

### George Jones

Staff Emeritus
No.

For $x < 0$, what is $d \left| x \right| /dx$?

For $x > 0$, what is $d \left| x \right| /dx$?

3. Jan 19, 2010

### Char. Limit

For x<0, the derivative is -1. For x>0, the derivative is +1.

Oh, that's not the unit step function, is it? Is there a "function" with those properties?

4. Jan 19, 2010

### cronxeh

5. Jan 19, 2010

### jgens

The sign function probably comes the closest but it's defined at $x = 0$ while the derivative of $|x|$ is not. But if you're interested, here's a brief discussion about it: http://en.wikipedia.org/wiki/Sign_function

6. Jan 19, 2010

### Char. Limit

I am interested.

Thanks for the help. I figured that the derivative was either abs(x)/x or x/abs(x), but I couldn't remember if such a function had a name. All I could think of was the unit step function (it looked like a step to me).

Does the sign function have a great use in mathematics other than being interesting?