What's the notation of the one-sided differential of a function? Assume
$$f(x)=\left \{\begin{matrix}
x-1 & x \leq 0\\ 
x+1 & x > 0
\end{matrix}  \right.$$
Obviously, $df(x)/dx$ do not exists at $x=0$ but the one-sided differential exists.
What's the notation of one-sided differential?
$$d_-f(x)/d_-x$$, $$d_-f(x)/dx$$, or $$df(x_-)/dx_-$$?
Update:
The previous example is really bad, and I was confusing deritive with differential. Thank you very much for the anwsers. Maybe the following example will make the problem clearer:
$$z=\left \{\begin{matrix}
y\times x & x \leq 0\\ 
2\times y \times x & x > 0
\end{matrix}  \right.$$
and both a and x is variable. 
How do I represent $\partial z / \partial x$ at $x=0^+$ and $x=0^-$?
It seems $\lim_{x\to 0^+}\partial z / \partial x$ and $\lim_{x\to 0^-}\partial y / \partial x$ is what I want.
 A: Seems the question is about left/right derivative.
The notation is normally: $f'(x^+)$ and $f'(x^-)$.
In your case it would be: $f'(0^+)$ and $f'(0^-)$.      
A: I am not sure what you mean by one-sided derivatives especially when you say that they exist.
According to wikipedia, the notation for left and right derivative of $f(x)$ at $x=a$ is
$$\partial_+f(a)=\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}$$
$$\partial_-f(a)=\lim_{x\to a^-}\frac{f(x)-f(a)}{x-a}$$
According to that definition, for your function
$$\partial_-f(0)=1$$
But
$$\partial_+f(0)$$ does not exists.
I agree with Hans Lundmark that alternative and more common notations may be $f'_-(a)$ and $f'_+(a)$.
As for
$$\lim_{x\to a^+}f'(x)$$
and
$$\lim_{x\to a^-}f'(x)$$
which need not be equal to the left- right- derivatives. I don't know any common notation for them. May be we have to put up with this and write
$$\lim_{x\to a^+}f'(x)$$
and
$$\lim_{x\to a^-}f'(x)$$
to avoid ambiguity.
In your example
$$\lim_{x\to0^-}f'(x)=\lim_{x\to0^+}f'(x)=1$$
