Thursday, November 17, 2011

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

No comments:

Post a Comment