Formula for Repeated Derivatives

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.