equation solving - Replace function and its derivatives found with Solve

If I have an equation

eq = f[x] + g[x] == 0

and I solve for f[x]

sol = Solve[eq,f[x]]

how do I use sol to replace both f[x] and its derivatives in another equation? For example

eq2 = f'[x]*g[x] + f[x] == 0

and then

eq2 /. sol

outputs

{-g[x] + g[x] f'[x] == 0}

Answer

The problem is caused by the internal structure of Derivative:

f'[x] // FullForm

(* ==> Derivative[1][f][x]  *)

There's no f[x] in there to replace. Using D[f[x], x] instead of f'[x] and Holding the whole equation works:

eq2 = Hold[D[f[x], x]*g[x] + f[x] == 0];


ReleaseHold[eq2 /. sol]

(*  ==> {-g[x] - g[x] Derivative[1][g][x] == 0}  *)

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