# `∂` — Derivative / VJP

Monadic operator. Keys: `Alt-f`

`f∂X`: gradient of `f` at `X`. `f` must return a unit. `U f∂ X`:
vector–Jacobian product, where `U` has the shape of `f`’s result. The
result has the shape of `X`.

Currently differentiates real evaluators `C⌻`, including factored and
exponent-table forms, and functions with a derivative that `f∂←g`
attaches. `C⌻` is a train that binds `C`, the same as `C↣⌻`.

``` bpl
f←[1 2 3]ₓ⌻ ⋄ f∂2ₓ                ⍝ 14ₓ
f←[1 2 3]ₓ⌻ ⋄ f∂∂2ₓ               ⍝ 6ₓ
f←[1 2 3]ₓ⌻ ⋄ [10 20]ₓ f∂ [1 2]ₓ  ⍝ [80 280]ₓ
```

Multivariate gradients retain the coordinate enclosure. A shared unit
coordinate sums partials.

``` bpl
f←⊂[1 2 0⋄1 0 2]ₓ ⌻ ⋄ f∂⊂[3 4]ₓ      ⍝ ⊂[6 8]ₓ
f←⊂[1 2 0⋄1 0 2]ₓ ⌻ ⋄ f∂3ₓ           ⍝ 12ₓ
```

Repeated `∂` requires unit input and output and one variable.

`f∂←g` attaches `g` as the derivative of the function that `f` names, as
[assignment](../assignment.ipynb#inverses-and-derivatives) describes.
`f` still applies itself. `f∂X` then calls `g X`, and `U f∂ X` calls
`U g X`, so `g` gives the gradient with one argument and the
vector–Jacobian product with two. A higher order differentiates `g`, and
`⁻¹` inverts `f`. [`⇄`](right-left-arrows.qmd) follows the derivative of
a function that has one.

``` bpl
r←{⍵×⍵} ⋄ r∂←{⍺←1 ⋄ ⍺×2×⍵}
r 3        ⍝ 9
r∂3        ⍝ 6
10 r∂ 3    ⍝ 60
```

## Errors

- `RANK`: monadic array output
- `LENGTH`: wrong cotangent shape
- `DOMAIN`: unsupported function or nonfinite/non-real data