Does there exist a formula akin to Cauchy's Repeated Integration formula, but for derivatives?
Cauchy's formula doesn't seem ideal for finding, say, the 100th derivative of a function as factorials are not defined for negative integers.
If such a formula exists, does it extend to reals (as in fractional derivatives)?
Thank you.
To differentiate an integer amount of times, one can simply differentiate repeatedly. To differentiate a non-integer amount of times, Cauchy's formula for repeated integration may be used to handle the non-integer part before differentiating.
For example, the $99.3$th derivative of $f(x)$ can be defined as
$$_aD_x^{99.3}f(x)=\frac1{\Gamma(0.7)}\frac{\mathrm d^{100}}{\mathrm dx^{100}}\int_a^x(x-t)^{-0.3}f(t)~\mathrm dt$$
or in other words, the $100$th derivative of the $0.7$th integral of $f$.
This is known as the Riemann-Liouville fractional derivative.