Infinite derivative

Solution 1:

$\frac{d^{\infty}}{dx^{\infty}}$ does not have a generally accepted meaning, as far as I know. But a reasonable definition would be

$$\frac{d^{\infty}}{dx^{\infty}}f(x) = \lim_{n\to \infty}\frac{d^n}{dx^n}f(x)$$

which only makes sense if this limit exists. Certainly it exists for some functions: all polynomials $p(x)$, for instance, with

$$\frac{d^{\infty}}{dx^{\infty}}p(x) = 0$$

And the exponential function:

$$\frac{d^{\infty}}{dx^{\infty}}e^x = e^x$$

And for some functions, the limit exists for some but not all values of $x$. For instance,

$$\frac{d^{\infty}}{dx^{\infty}}e^{-x^2} = 0$$

at $x=0$, but is not well-defined for any other value of $x$.

So it's not a question of whether this "infinite derivative" is possible; but of how it should be defined. Another question is whether such a concept is useful; I don't think the definition I have suggested here is any use, but I might be wrong.

Solution 2:

A functions $f$ is said to be of derivability class $C^k$ if its derivatives $f',f'',f''',f^{(k)}$ exist and are continous, a function that has derivatives of every order is said to be of class $C^{\infty}$, searching for derivability classes will give you more informations on this topic.

Solution 3:

This answer is in the spirit of the answer of TonyK. He considers a sequence of functions composed by classical derived functions and defines its limit, provided it existed, as the infinite-th derivative.

Alternatively one could use a family of functions composed by fractional derivatives (provided they exist) and take a limit of that family as a candidate for the infinite-th derivative. One way to define a fractional derivative (cf. https://en.wikipedia.org/wiki/Fractional_calculus) as an operator is via integral transforms depending on a parameter $n$ describing the order. An idea to proceed to the infinite-th derivative may be to take a limit of such a family of functions or correspondingly a family of operators.

Probably this requires some hypothesis on the original function to be well-defined. It is not clear (to me) if this approach is giving anything useful or meaningful and if it does whether it yields anything different from the approach of TonyK. It might however be worth giving it some thoughts.

Disclaimer: I am not an analyst.