Modifying a single index of a vector valued function doesn't work

[Nidish96]

The following works:

N = 8;
t = (0:N-1)*2π/N;
yt = cos.(t);
fun = (yt) -> begin yf = rfft(yt); yout = real(yf); return yout; end;
Dh = ForwardDiff.jacobian(fun, yt)

while the following doesn't:

N = 8;
t = (0:N-1)*2π/N;
yt = cos.(t);
fun = (yt) -> begin yf = rfft(yt); yout = real(yf); yout[1] .*= 2; return yout; end;
Dh = ForwardDiff.jacobian(fun, yt)

Curiously enough, the following works:

N = 8;
t = (0:N-1)*2π/N;
yt = cos.(t);
fun = (yt) -> begin yf = rfft(yt); yout = real(yf); yout[1] = 2yout[1]; return yout; end;
Dh = ForwardDiff.jacobian(fun, yt)

I'd appreciate any leads!

[KristofferC]

None of these work for me:

julia> Dh = ForwardDiff.jacobian(fun, yt)
ERROR: type ForwardDiff.Dual{ForwardDiff.Tag{var"#11#12", Float64}, Float64, 8} not supported
Stacktrace:
  [1] error(s::String)
    @ Base ./error.jl:35
  [2] _fftfloat(::Type{ForwardDiff.Dual{ForwardDiff.Tag{var"#11#12", Float64}, Float64, 8}})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:38
  [3] _fftfloat(x::ForwardDiff.Dual{ForwardDiff.Tag{var"#11#12", Float64}, Float64, 8})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:39
  [4] fftfloat(x::ForwardDiff.Dual{ForwardDiff.Tag{var"#11#12", Float64}, Float64, 8})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:34
  [5] realfloat(x::Vector{ForwardDiff.Dual{ForwardDiff.Tag{var"#11#12", Float64}, Float64, 8}})
    @ AbstractFFTs ~/.julia/packages/AbstractFFTs/4iQz5/src/definitions.jl:49
  [6] plan_rfft(x::Vector{ForwardDiff.Dual{ForwardDiff.Tag{…}, Float64, 8}}, region::U