Derivative of |x|: Unit Step Fn Equal?

[Char. Limit]

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

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

 

[George Jones]

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

No.

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

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

 

[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?

 

[cronxeh]

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?

http://en.wikipedia.org/wiki/Signum_function

 

[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

 

[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?