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Derivative | m D. n u D. n mu |

`u D. n` is the `n`-th derivative of`u`,`` and`u`v D. n` is`u` with an assigned`n`-th derivative`v` .`` For example:

       (cube D.1;cube D.2; (cube=: ^&3"0) D.3)y=: 2 3 4
    +--------+--------+-----+
    |12 27 48|12 18 24|6 6 6|
    +--------+--------+-----+

The derivative applies to constant functions, polynomials, the exponential`^`,`` the integral powers`^&n`,`` and those assigned by`u`v D. n` .`` It also applies to functions derived from these by addition, subtraction, multiplication, and division`` (`u+v`, etc.); by the composition`u@v`;`` and by the inverse`u^:_1` .`` Since functions such as`j.` and`-` (negation) and`%:` (square root) and`1&o.` (sin) and`6&o.` (cosh) may all be so derived, they are also in the domain of the derivative. Others are treated by approximation. The derivative of an arbitrary function may also be treated by a polynomial approximation, (provided by the matrix divide), or by approximations using the secant slope`D:` .
If the argument rank of`u` is`a` and the result rank is`r`,`` then the argument rank of`u D.1` is also`a`,`` but its result rank is`r+a` :`` the result of`u D.1` is the derivative of each atom of the result of `u` with respect to each element of its argument, giving what is commonly referred to as the partial derivatives. For example:

       volume=: */"1
       VOLUMES=: */\"1
       (volume;volume D.1;VOLUMES;VOLUMES D.1) y
    +--+------+------+------+
    |24|12 8 6|2 6 24|1 3 12|
    |  |      |      |0 2  8|
    |  |      |      |0 0  6|
    +--+------+------+------+

       determinant=: -/ . *
       permanent=: +/ . *
       (];(determinant D.1);(permanent D.1))m=:*:i.3 3
    +--------+--------------+--------------+
    | 0  1  4|_201  324 _135|2249 1476 1017|
    | 9 16 25| 132 _144   36| 260  144   36|
    |36 49 64| _39   36   _9|  89   36    9|
    +--------+--------------+--------------+

The adverbs`D=: 1 : 'x."0 D.1'` and`VD=: 1 : 'x."1 D.1'` assign ranks to their arguments, then take the first derivative; they are convenient for use in scalar and vector calculus:

       sin=: 1&o.
       x=: 0.5p1 _0.25p1
       (*/\ VD y);(sin x);(sin D x);(sin D D x)
    +------+-----------+----------+-----------+
    |1 3 12|1 _0.707107|0 0.707107|_1 0.707107|
    |0 2  8|           |          |           |
    |0 0  6|           |          |           |
    +------+-----------+----------+-----------+

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