Generalized Derivative: Denote $d_U,l,f(x)$   as the expression:
$$\lim:  \delta \rightarrow l, U(f(x), \delta )  $$
It's trivial to show that the standard derivative is simply a case of $l = 0$ and $U = \dfrac{f(x + \delta) - f(x)}{\delta}$
Are there other flavors of derivatives yielded by this notation? Can one then create different orders of calculus depending on the type of derivative utilized and create a general theory of Calculuses? 
Has this been done before and if so where should I look for more information?
 A: If $l \not=0$, then your limit can be evaluated just by plugging in $l$:
$$\lim\limits_{\delta \to l} U(f(x),\delta) = \lim\limits_{\delta \to l} \frac{f(x+\delta)-f(x)}{\delta} = \frac{f(x+l)-f(x)}{l}$$
We need the limit only when $l=0$. 
Has this been studied? Well, yes, it's a difference quotient. $U(f(x),l)$ measures the average rate of change moving from $(x,f(x))$ to $(l,f(x+l))$.
Addendum: Maybe this is what you're looking for...
What if we consider something like: 
$$\lim\limits_{h \to 0} \frac{f(x+h^2)-f(x)}{h}$$
Would this give us an interesting new theory? 
Well, no. Here's the idea of why it doesn't:
$f(x+h^2) \approx f(x) + f'(x)h^2 + f''(x)/2 h^4 + \cdots$ so  $(f(x+h^2)-f(x))/h = (f'(x)h^2+f''(x)/2 h^4 + \cdots)/h = f'(x)h + f''(x)/2 h^3 + \cdots$ (whose limit is zero).
In general if you try to cook up some "new" difference quotient, you'll either get something like this above (always zero so very uninteresting) OR you'll get something that's essentially just a regular derivative anyway.
