For integer $n$ we have that: $$\frac{d^n}{dx^n} \sin(x) = \sin\left(x+\frac{n \pi}{2}\right)$$ For any function $f(x)$ (ignoring domain restrictions for the time being), let: $$f(x) = \sin(u) \ \ [1]$$ $$u = \arcsin\left(f(x)\right) \ \ [2]$$ Then, take the $n$th derivative wrt $u$ of both sides of $[1]$: $$\frac{d^n}{du^n} f(x) = \frac{d^n}{du^n} \sin(u)$$ $$\frac{d^n}{du^n} f(x) = \sin\left(u+\frac{n \pi}{2}\right)$$ Now we can solve for $du$ and substitute: $$f'(x) \ dx = \cos(u) \ du$$ $$du = \frac{f'(x)}{\cos(u)} \ dx = \frac{f'(x)}{\cos\left(\arcsin(f(x)\right)} \ dx = \frac{f'(x)}{\sqrt{1-f(x)^2}} \ dx $$
Then: $$\frac{d^n}{du^n} f(x) = \frac{d^n}{\left(\frac{f'(x)}{\sqrt{1-f(x)^2}} \ dx \right)^n} \ f(x) = \left(\frac{f'(x)}{\sqrt{1-f(x)^2}}\right)^{-n} \frac{d^n}{dx^n} f(x)$$
Finally we can solve for the $n$th derivative in terms of $x$ and $n$ alone, using $[2]$: $$\frac{d^n}{dx^n} f(x) = \sin\left(\arcsin\left(f(x)\right)+\frac{n \pi}{2}\right) \left(\frac{f'(x)}{\sqrt{1-f(x)^2}}\right)^{n}$$
And now there is no issue with letting $n$ vary continuously. Now from this formula we have that $|f(x)| < 1$, so it obviously doesn't work everywhere for all functions. I made a Desmos graph, try inputting different choices of $f$ and see how it interpolates between the function and its first derivative.
My questions are the following:
- Why is there sometimes not a smooth transition between the function and its derivative? Try letting $f = x^{x}$ and see what happens when you let $n$ go from $0$ to $1$, the left half flips back and forth. Is this due to the exponentiation not being defined?
- How do we extend the domain? We can just make the transformation $f(x) \to f(x - c)$ to center the derivative around a point other than 0, but it's not clear how to make this accept functions with a magnitude greater than $1$.
- Is this always exact for integer n so long as $|f(x)| < 1$? It seems like there might be a bit of error, but can't tell if that is numerical or due to the domain of the functions involved.
- Has anything like this been done before? I'd like to see if there is any existing literature that uses this sort of idea.
I've managed to extend the domain by simply putting $f(x)=A\sin{u}$, where $A$ is just a really large number.
Then you can get the final equation like this:
$$ \frac{d^n}{dx^n} f(x) = A\sin\left(\arcsin\left(\frac{f(x)}{A}\right)+\frac{n \pi}{2}\right) \left(\frac{f'(x)}{\sqrt{A^2-f(x)^2}}\right)^{n} $$
I also plotted it as a Desmos Graph,
and according to my calculations, it should be extend to $|x|=A$, but it stops around $|x|=\sqrt{A}$.
I think this happened because you ignored the domain restrictions during the calculation.