Chain rule fails on real, imag, conj, with non-real symbolic functions

[amilsted]

julia> using Symbolics

julia> using SymbolicUtils

julia> @syms x::Real f(::Real)::Number
(x, f)

julia> Symbolics.derivative(cos(f(x)), x)    # so far so good
-Differential(x)(f(x))*sin(f(x))

julia> Symbolics.derivative(conj(f(x)), x)   # Symbolics doesn't know how to differentiate these,
Differential(x)(conj(f(x)))*Differential(x)(f(x))

julia> Symbolics.derivative(imag(f(x)), x)   # and it's so upset
Differential(x)(imag(f(x)))*Differential(x)(f(x))

julia> Symbolics.derivative(real(f(x)), x)   # it gets the chain rule wrong
Differential(x)(real(f(x)))*Differential(x)(f(x))

Btw, I am using Numberfor f here to represent a function that might be complex. Same thing happens for f(::Real)::Complex. I am not using Complex{Num} because I do not want to split real and imaginary parts.

[amilsted]

Relatedly, it would be very nice if we could add rules for D(x)(conj(f(x))) -> conj(D(x)(f(x))) and similar for complex f but real x.