I have been wondering whether the following limit is being used somehow, as a variation of the derivative:
$$\lim_{h\to 0} \frac{f(x+h)-f(x-h)}{2h} .$$
Edit: I know that this limit is defined in some places where the derivative is not defined, but it gives us some useful information.
The question is not whether this limit is similar to the derivative, but whether it is useful somehow.
Thanks.