Finding Multiple Derivatives

      When finding derivatives, it becomes tedious to continually derive the function to find further derivatives.  Although I'm not sure of the application of derivatives past the second, I came up with a formula for multiple derivations of polynomials.  To do this I observed the way each part of a term changes over each differentiation.

      Each time a new derivative is found, the coefficient is multiplied by the power.  Thus:

            C₁ = C₀.n(n - 1)(n - 2) ... (n - r) for all n > r

                 = C₀.ⁿP_r

      where C₀ is the original coefficient, n gives the original power of x, and r gives the derivative to be found (e.g. For f'''(x), r = 3)

      Each time a derivative is found, the constant term is lost.  Thus:
           C₁ = 0, for all 0 < n < r



      For negative indices, the sign of the coefficient depends on whether r is odd or even.  The magnitude of the coefficient (absolute value) is multiplied by a larger number each time a new derivative is found.  Therefore, beginning multiplication at the original power will give the new coefficient as follows:



           C₁ = C₀.(-1)^r.|n - r + 1|! / |n + 1|!, for all n < 0



      As the powers are simply lessened, in magnitude, by 1 for each derivative, and those for which n is positive and less than r should be eliminated, they are given by:
           n₁ = n₀ - r, for all n₀  > r

           n₁ = 0, for all 0 < n₀ < r     (as x⁰ = 1, and the coefficient is 0, thus term is 0 x 1 = 0)

           n₁ = n₀ + r, for all n₀ < 0

      Collating these results gives a way of finding a term in the derivative of a polynomial function:

      Where T_k = C.xⁿ for a polynomial function f(x).  t_k in the r^th derivative of f(x) is given by:

             t_k = C.ⁿP_rx^{n - r}, for all n > r

             t_k = 0, for all 0 < n < r
             t_k = C.(-1)^r.|n - r + 1|! / |n + 1|!.x^{n + r}, for all n < 0