partial at one coordinate

partial.jl

julia> using StructArrays

julia> using ForwardDiff

julia> function partiali(n,i)
           ith = zeros(n)
           ith[i] += 1
           function (f,x)
               sa = StructArray{ForwardDiff.Dual{}}((x, ith))
               return f(sa)
           end
       end
partiali (generic function with 2 methods)

julia> f(x) = sum(x -> x^2, x)
f (generic function with 1 method)

julia> x = randn(1000);

julia> ∂₅₀ = partiali(1000,50);

julia> x[50]
2.1129129387789667

julia> @btime $f($x)
  82.987 ns (0 allocations: 0 bytes)
1000.4848272576928

julia> @btime $∂₅₀($f,$x)
  167.038 ns (0 allocations: 0 bytes)
Dual{Nothing}(1000.4848272576928,4.225825877557933)