`DiffRules.diffrule`

— Method`diffrule(M::Union{Expr,Symbol}, f::Symbol, args...)`

Return the derivative expression for `M.f`

at the given argument(s), with the argument(s) interpolated into the returned expression.

In the `n`

-ary case, an `n`

-tuple of expressions will be returned where the `i`

th expression is the derivative of `f`

w.r.t the `i`

th argument.

**Examples**

```
julia> DiffRules.diffrule(:Base, :sin, 1)
:(cos(1))
julia> DiffRules.diffrule(:Base, :sin, :x)
:(cos(x))
julia> DiffRules.diffrule(:Base, :sin, :(x * y^2))
:(cos(x * y ^ 2))
```

`DiffRules.diffrules`

— Method`diffrules(; filter_modules=(:Base, :SpecialFunctions, :NaNMath))`

Return a list of keys that can be used to access all defined differentiation rules for modules in `filter_modules`

.

Each key is of the form `(M::Symbol, f::Symbol, arity::Int)`

. Here, `arity`

refers to the number of arguments accepted by `f`

and `M`

is one of the modules in `filter_modules`

.

To include all rules, specify `filter_modules = nothing`

.

Calling `diffrules()`

with the implicit default keyword argument `filter_modules`

does *not* return all rules defined by this package but rather only rules for the packages for which DiffRules 1.0 provided rules. This is done in order to not to break downstream packages that assumed this list would never change. It is planned to change `diffrules()`

to return all rules, i.e., to use the default keyword argument `filter_modules=nothing`

, in an upcoming breaking release of DiffRules.

**Examples**

```
julia> first(DiffRules.diffrules()) isa Tuple{Symbol,Symbol,Int}
true
julia> (:Base, :log, 1) in DiffRules.diffrules()
true
julia> (:Base, :*, 2) in DiffRules.diffrules()
true
```

If you call `diffrules()`

, only rules for Base, SpecialFunctions, and NaNMath are returned but no rules for LogExpFunctions:

```
julia> any(M === :LogExpFunctions for (M, _, _) in DiffRules.diffrules())
false
```

If you set `filter_modules=nothing`

, all rules defined in DiffRules are returned and in particular also rules for LogExpFunctions:

```
julia> any(
M === :LogExpFunctions
for (M, _, _) in DiffRules.diffrules(; filter_modules=nothing)
)
true
```

If you set `filter_modules=(:Base,)`

only rules for functions in Base are returned:

```
julia> all(M === :Base for (M, _, _) in DiffRules.diffrules(; filter_modules=(:Base,)))
true
```

`DiffRules.hasdiffrule`

— Method`hasdiffrule(M::Union{Expr,Symbol}, f::Symbol, arity::Int)`

Return `true`

if a differentiation rule is defined for `M.f`

and `arity`

, or return `false`

otherwise.

Here, `arity`

refers to the number of arguments accepted by `f`

.

**Examples**

```
julia> DiffRules.hasdiffrule(:Base, :sin, 1)
true
julia> DiffRules.hasdiffrule(:Base, :sin, 2)
false
julia> DiffRules.hasdiffrule(:Base, :-, 1)
true
julia> DiffRules.hasdiffrule(:Base, :-, 2)
true
julia> DiffRules.hasdiffrule(:Base, :-, 3)
false
```

`DiffRules.@define_diffrule`

— Macro```
@define_diffrule M.f(x) = :(df_dx($x))
@define_diffrule M.f(x, y) = :(df_dx($x, $y)), :(df_dy($x, $y))
⋮
```

Define a new differentiation rule for the function `M.f`

and the given arguments, which should be treated as bindings to Julia expressions. Return the defined rule's key.

The LHS should be a function call with a non-splatted argument list, and the RHS should be the derivative expression, or in the `n`

-ary case, an `n`

-tuple of expressions where the `i`

th expression is the derivative of `f`

w.r.t the `i`

th argument. Arguments should be interpolated wherever they are used on the RHS.

Note that differentiation rules are purely symbolic, so no type annotations should be used.

**Examples**

```
@define_diffrule Base.cos(x) = :(-sin($x))
@define_diffrule Base.:/(x, y) = :(inv($y)), :(-$x / ($y^2))
@define_diffrule Base.polygamma(m, x) = :NaN, :(polygamma($m + 1, $x))
```