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 functions
other 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 nth 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.