D: Secant Slope

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| u D: n mu mu | Secant Slope

|  | `x u D: 1 y` is the secant slope of the function`u` through the points`y` and`y+x` .`` The secant slope is generalized to the case`x u D. n y` in the manner of the derivative `D.` .`` The argument `x` may be a list, giving several slopes. In the general case, each item of`x` has the shape`{. $$"r y`, where`r` is the rank of`u`,`` therefore specifying the increment in each possible direction. An argument`x` of lower rank is extended in the usual manner. For example,`x=: 1e_8` provides the same increment in each direction and, because of the small magnitude, yields an approximation to the derivative.

       1 log D:1 y
    0.405465 0.287682 0.223144

       incr=: 1 0.1 0.01 1e_8
       incr log D:1/y
    0.405465 0.287682 0.223144
    0.487902 0.327898 0.246926
    0.498754 0.332779 0.249688
         0.5 0.333333     0.25

       log D.1 y
    0.5 0.333333 0.25
       %y
    0.5 0.333333 0.25

       f=: +/@:*:"1
       g=: +/@:*:\"1
       (f y) ; (1 f D:1 y) ; (1 0.1 1e_8 f D:1 y)
    +--+-----+--------------+
    |29|5 7 9|  5  0.61 8e_8|
    |  |     | 50   6.1 8e_7|
    |  |     |5e8 6.1e7    8|
    +--+-----+--------------+

       (g y) ; (1 g D:1 y)
    +-------+-----+
    |4 13 29|5 0 0|
    |       |5 7 0|
    |       |5 7 9|
    +-------+-----+

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