If f(x) = C.√xn, the rth derivative is given by:
For n > 2r, f(r)(x) = (C.n!!)/((n - 2r)!!2r).xn - 2r
For 0 < n < 2r, If n = 1 + 4k,
then f(r)(x) = (C.(-1)r + 1.n!!|n - 2r + 2|!!)/(2r).xn - 2
If n = 3 + 4k,
then f(r)(x) = (C.(-1)r.n!!|n - 2r + 2|!!)/(2r).xn - 2
For n < 0, f(r)(x) = (C.(-1)r.n!!|n - 2r + 2|!!)/(2r|n|!!).xn - 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)ar).xn - ar
For n > ar, f
For n < 0, f (r)(x) = (C.(-1)r.|n - ar + a|!(a))/(ar).xn - ar
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