# Vocabulary/ddot

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`u d. n y`Ordinary Derivative Conjunction

Rank 0 *-- operates on individual atoms of y, producing a result of the same shape --*
WHY IS THIS IMPORTANT?

The *ordinary* `n`*-th derivative* of [the mathematical function implemented by] verb `u`.

^. d. 1 NB. derivative of ln(x) is 1/x % ^.@(1&o.) d. 1 NB. derivative of ln(sin x) is cos x * 1/sin x 2&o. * %@(1&o.) (^&2 + ^&3) d. 1 NB. 1st derivative of x^2 + x^3 0 2 3x&p. (^&2 + ^&3) d. 2 NB. 2nd derivative of x^2 + x^3 2 6x&p.

Since `u d. n` concerns itself with **ordinary** rather than **partial** derivatives, `u` should be a verb with rank 0, and ` u d. n ` will then have rank 0.

### Common Uses

Doing Calculus.

### Related Primitives

Derivative (`u D. n`)

### More Information

1. The verb ` u d. n ` is meaningful only when used monadically.

2. `u` must be one of the verbs, or combinations of verbs, for which J knows the derivative. These are:

**Allowable forms of**`u`**in**`u d. 1`**Type****Allowed Values**constants `_9:`through`9:``_: m"0`monads `<: >: +: *: - -. -: % %: ^ ^. [ ] j. o. r.`bonded dyads `m&+ m&* m&- m&% m&%: m&^ m&^. m&! m&p. +&n *&n -&n %&n ^&n ^.&n`circle functions `0&o.`(`-.&.*:`),`1&o.`(sin),`2&o.`(cos),`3&o.`(tan),`5&o.`(sinh),`6&o.`(cosh),`7&o.`(tanh)inverses of the above for all monads; for bonded dyads **except**`m&! m&p. ^.&n`; for**no**circle functionsother inverses `m&j.^:_1 m&r.^:_1 %:&n^:_1 j.&n^:_1 r.&n^:_1`compounds where

`u`and`v`are allowed`u@v u@:v u&v u&:v (u + v) (u * v) (u - v) (u % v) (u , v)`rank `"n`allowed and ignored

3. `n` may be negative to calculate the *n*th antiderivative (with the constant of integration equal to 0).
The allowed forms of `u` are the same as for ` u d. 1 `, except that ` m&%: m&^. ^.&n m&^^:_1 %:&n^:_1 (u * u) (u % u) (u , u) ` are not allowed.

4. ` u d. n ` integrates symbolically rather than numerically, and should be used rather than Derivative (`D.`) where possible.

### Details

1. `n` may be a list, in which case the result for each atom of `y` will be the list of derivatives of orders `n`.