Is there a way to make sense of the nth derivative of a function when n is not a positive integer?
The notation f(n) is usually introduced in calculus classes in order to make Taylor’s theorem easier to state:
To make the above statement work, the 0th derivative is defined to be the function itself, i.e. don’t take any derivatives. This makes a modest extension of the notation f(n) from requiring n to be a positive integer to being a non-negative integer. Can we make sense of the case when n is negative or non-integer? Before answering that question, let’s think about what the fractional derivative might be in some special cases if we could define it.
When we take derivatives of powers of x, we get factorial-like coefficients. The first derivative of xm is m xm-1, the second derivative is m(m-1) xm-2, the third derivative is m(m-1)(m-2) xm-3, etc. We can use the gamma function to extend the factorial function to non-integer argument, so maybe we could do the same to compute non-integer derivatives. If m > -1, and n is a positive integer, the nth derivative of xm is (m!/(m-n)!) xm-n. We could rewrite this as (Γ(m+1) / Γ(m-n+1)) xm-n. The result holds for integer values of n, and so we could hope it holds for non-integer values of n.
If n is an integer and we take the nth derivative of ebx we get bn ebx. We might guess that for non-integer values of n the same formula holds.
It is indeed possible to define derivatives of order n for non-integer values of n, and the speculations above are correct, subject to some conditions. In fact there are several ways to define non-integer derivatives and the differences can be complicated.
What about negative derivatives? Well, it makes sense that these could be anti-derivatives, i.e. integrals. We could define, for example, the -3rd derivative of f(x) to be a function whose third derivative is f(x). However, anti-derivatives are only determined up to a constant. We could use the Fundamental Theorem of Calculus to uniquely specify anti-derivatives if we agree on a lower limit of integration c, such as c = 0 or maybe c = -∞.
Here’s one way fractional derivatives could be defined. Suppose the Fourier transform of f(x) is g(ξ). Then for positive integer n, the nth derivative of f(x) has Fourier transform (2π i ξ)n g(ξ). So you could take the nth derivative of f(x) as follows: take the Fourier transform, multiply by (2π i ξ)n, and take the inverse Fourier transform. This suggests the same procedure could be used to define the nth derivative when n is not an integer.
Fractional derivatives have practical uses. The book An Atlas of Functions makes frequent use of fractional derivatives, especially derivatives of order 1/2 and -1/2, to show connections between different classes of functions.
Related article: Generalizing binomial coefficients