Higher Derivatives for Fractional Powers

      This is an extension to my work on "Finding Multiple Derivatives," which deals with polynomials.  To find these, I split the eventual term up into three independent parts: the sign, the numerator, and the denominator.  Firstly I began by looking at the derivatives for which the power is n/2 (i.e. the function is a constant multiplied by the square root xⁿ).  This brought separate formulas for three domains of n and two sub-situations for when n is between 0 and twice the number of derivatives being found (0 < n < 2r).  Considering these brought the following:

        If f(x) = C.√xⁿ, the r^th derivative is given by:

                For n > 2r, f^(r)(x) = (C.n!!)/((n - 2r)!!2^r).x^{n - 2r}

                    For 0 < n < 2r, If n = 1 + 4k,

                        then f^(r)(x) = (C.(-1)^{r + 1}.n!!|n - 2r + 2|!!)/(2^r).x^{n - 2}

                                        If n = 3 + 4k,

                        then f^(r)(x) = (C.(-1)^r.n!!|n - 2r + 2|!!)/(2^r).x^{n - 2}

                For n < 0, f^(r)(x) = (C.(-1)^r.n!!|n - 2r + 2|!!)/(2^r|n|!!).x^{n - 2}

       where k is an integer

       From here I determined a couple of formulae for similar situations, in a more general form.  However, I have not yet found a solid general formula for powers between 0 and the number root multiplied by the derivative being found (ie. 0 < n < ar for the rth derivative of C.^a√xⁿ).  General derivatives for fractional powers of x can be found by considering:
      
      For n > ar, f^(r)(x) = (C.n!^(a))/((n - ar)!^(a)a^r).x^{n - ar}

      For n < 0, f^(r)(x) = (C.(-1)^r.|n - ar + a|!^(a))/(a^r).x^{n - ar}