p.. Poly. Deriv. - Poly. Integral

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Polynomial Derivative | p.. 1 0 1 | Polynomial Integral

Applied to a polynomial (coefficients or boxed roots),`p..` produces the coefficients of the derivative of polynomial. For example:
       p.. 1 2 3 4 5
    2 6 12 20
       p.. 5 4 3 2 1
    4 6 6 4

       p.. 2; 1j1 1j_1
    _4 4
       p.. p. 2; 1j1 1j_1
    _4 4
|  | `x p.. y` produces the integral of polynomial`y` with a constant term of`x` .`` Thus:
       5 p.. 4 6 6 4
    5 4 3 2 1
       1 p.. 2 6 12 20
    1 2 3 4 5

Further examples:

       p.. 1&o. t. i. 11x              NB. derivative of sine
    1 0 _1r2 0 1r24 0 _1r720 0 1r40320 0
       2&o. t. i.10x                   NB. cosine
    1 0 _1r2 0 1r24 0 _1r720 0 1r40320 0

       p.. 2&o. t. i. 11x              NB. derivative of cosine
    0 _1 0 1r6 0 _1r120 0 1r5040 0 _1r362880
       -@(1&o.) t. i.10x               NB. minus sine
    0 _1 0 1r6 0 _1r120 0 1r5040 0 _1r362880
       - (1&o. t. i.10x)
    0 _1 0 1r6 0 _1r120 0 1r5040 0 _1r362880

       p..^:(i.@#) 8 $ 1
       1    1    1   1   1  1 1 1
       1    2    3   4   5  6 7 0
       2    6   12  20  30 42 0 0
       6   24   60 120 210  0 0 0
      24  120  360 840   0  0 0 0
     120  720 2520   0   0  0 0 0
     720 5040    0   0   0  0 0 0
    5040    0    0   0   0  0 0 0

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