# Is it possible to apply rules on Differentials in Symbolics.jl?
**URL:** https://discourse.julialang.org/t/is-it-possible-to-apply-rules-on-differentials-in-symbolics-jl/134936
**Category:** General Usage
**Tags:** symbolics
**Created:** [January 7, 2026, 2:37pm UTC](https://discourse.julialang.org/t/is-it-possible-to-apply-rules-on-differentials-in-symbolics-jl/134936 "2026-01-07T14:37:51Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@AwesomeQuest](https://discourse.julialang.org/u/AwesomeQuest)
#### Post date: [January 7, 2026, 2:37pm UTC](https://discourse.julialang.org/t/is-it-possible-to-apply-rules-on-differentials-in-symbolics-jl/134936/1 "2026-01-07T14:37:51Z")
I’m trying to apply the chain rule to effectively do a change of variables but I can’t figure out a rule that works.
Here is an example that I think should work.
```julia-auto
@variables a b c(a,b)
r = @rule Differential(a)(~x) => Differential(a)(b)*Differential(b)(~x)
r(D(a)(c)) # nothing
```
What am I doing wrong?
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [January 8, 2026, 3:59pm UTC](https://discourse.julialang.org/t/is-it-possible-to-apply-rules-on-differentials-in-symbolics-jl/134936/2 "2026-01-08T15:59:54Z")
That might need an issue
---
### Author:  [@AwesomeQuest](https://discourse.julialang.org/u/AwesomeQuest)
#### Post date: [January 8, 2026, 10:07pm UTC](https://discourse.julialang.org/t/is-it-possible-to-apply-rules-on-differentials-in-symbolics-jl/134936/3 "2026-01-08T22:07:31Z")
Related question:
Is there a faster way of generating the substitution dictionary that does the same thing as the rule above. Below I’ve included the code I’ve written to generate the substitution `Dict`. Does anyone see a faster way to make this dictionary?
```julia-auto
using Symbolics, Combinatorics
D = Differential
# The intention of this substitution code is
# to transform derivatives w.r.t. x of an unknown
# function ϕ into derivatives w.r.t. ξ
# The backward transform x(ξ) is known and so are its derivatives
# and dξ/dx is known as the inverse of the jacobian of x(ξ)
# Thus everything is to be written in terms of
# derivatives w.r.t. ξ of the functions ϕ and dξ/dx
N = 2
# x are the real coordinates
# ξ are the transformed coordinates
# dξdx is a workaround since (D(ξ[2])*D(x[1]))(ξ[1]) evaluates to 0
# dξdx[i,j] = dξ[j]/dx[i]
# ϕ is an unknown function
@variables x[1:N] (ξ(x...))[1:N] (dξdx(..))[1:N, 1:N] ϕ(..)
ϕ = ϕ(x...,ξ...)
dξdx = dξdx(ξ...,x...)
# The maximum number of derivative that
# the substitution dictionary should contain
maxdiff = 3
# This function is extreamly slow for N > 2
function generatesubsdict(N=N, maxdiff=maxdiff)
outputdict = Dict{Num,Num}()
# Generate all possible permutations of derivatives w.r.t. ξ
# since any one of them could appear in an equation
# differentiated w.r.t. x
xiperms = multiset_permutations.(
[repeat(1:N,maxdiff)], 1:maxdiff
) |> Iterators.flatten .|> x->x .|> x->ξ[x]
xidiffs = [prod(D.(xi)) for xi in xiperms]
# This loops over all xs
for i in 1:N
# Substitute all d/dx = sum(dξ/dx*d/dξ over ξ)
outputdict[D(x[i])(ϕ)] =
sum(dξdx[i,xi]*D(ξ[xi])(ϕ) for xi in 1:N)
# Same for all possible derivatives of ϕ w.r.t. ξ
for dxi in xidiffs
phidiff = dxi(ϕ)
outputdict[D(x[i])(phidiff)] =
sum(dξdx[i,xi]*D(ξ[xi])(phidiff)
for xi in 1:N)
end
# These are the cases of differentiating ξ w.r.t. x multiple times
# d/dx_i dξ_j/dx_k = sum((d/dξ_l dξ_j/dx_k)*dξ_l/dx_i over l)
for j in 1:N, k in 1:N
outputdict[D(x[i])(dξdx[j,k])] =
sum(D(ξ[xi])(dξdx[j,k])*dξdx[i,xi] for xi in 1:N)
for dxi in xidiffs
xidiff = dxi(dξdx[j,k])
outputdict[D(x[i])(xidiff)] =
sum(D(ξ[xi])(xidiff)*dξdx[i,xi]
for xi in 1:N)
end
end
end
return outputdict
end
```
---
### Author:  [@AwesomeQuest](https://discourse.julialang.org/u/AwesomeQuest)
#### Post date: [January 9, 2026, 10:01am UTC](https://discourse.julialang.org/t/is-it-possible-to-apply-rules-on-differentials-in-symbolics-jl/134936/4 "2026-01-09T10:01:08Z")
I’ve made an issue [Rules don't work on Differentials · Issue #1748 · JuliaSymbolics/Symbolics.jl · GitHub](https://github.com/JuliaSymbolics/Symbolics.jl/issues/1748#issue-3796298627)