Has this concept been explored & if so what name does it go by?
Taking a simple polynomial & its derivatives:
$$y = x^3 + x ^ 2 + x + 1$$ $$\frac{dy}{dx} = 3x^2 + 2x + 1$$ $$\frac{d^2y}{dx^2} = 6x + 2$$ $$\frac{d^3y}{dx^3} = 6$$
At a glance, differentiation seems like a discrete operation. However it looks like it cuold be continuous.
Let's say q is the degree of differentiation, and for normal highschool differentiation $q = 1$, and we now want to try differentiating where $0 < q < 1$.
The process of differentiating a component in a polynomial becomes something like:
$vx^u$ becomes $(v+qu(v-1))x^(u-q)$
When q = 1 we still get the first descrete derivative as $v+1u(v-1) = vu$, while when q = 0 we get the original formula as $v+0u(v-1) = v$, so we have a q representing a continuous variable between the two formulae. Dragging q as a slider on a graph confirms it toggles between the two:
https://www.desmos.com/calculator/eqitd8v37x
I'm interested to know:
- What name does this sort of operation go by?
- Are there smoother or more elegant ways to make make continuous paths between a function and its derivative?
- Is there a more general way which allows q > 1 to represent higher derivatives? (Factorials?)
- Are there more general operations allowing us to make continuous paths between other formulae? (Beyond just weighting & adding their outputs)
See http://en.wikipedia.org/wiki/Fractional_calculus - this may contain all that you are looking for.