DS-1000 / 53 / Sample dataframe:

prompt

Problem:
Sample dataframe:
df = pd.DataFrame({"A": [1, 2, 3], "B": [4, 5, 6]})

I'd like to add sigmoids of each existing column to the dataframe and name them based on existing column names with a prefix, e.g. sigmoid_A is an sigmoid of column A and so on.
The resulting dataframe should look like so:
result = pd.DataFrame({"A": [1, 2, 3], "B": [4, 5, 6], "sigmoid_A": [1/(1+e^(-1)), 1/(1+e^(-2)), 1/(1+e^(-3))], "sigmoid_B": [1/(1+e^(-4)), 1/(1+e^(-5)), 1/(1+e^(-6))]})

Notice that e is the natural constant.
Obviously there are redundant methods like doing this in a loop, but there should exist much more pythonic ways of doing it and after searching for some time I didn't find anything. I understand that this is most probably a duplicate; if so, please point me to an existing answer.

A:
<code>
import pandas as pd


df = pd.DataFrame({"A": [1, 2, 3], "B": [4, 5, 6]})
</code>
result = ... # put solution in this variable
BEGIN SOLUTION
<code>