A brief introduction to Functional Calculus
Let 
be a functional of the functions y(x) and z(x).
Then this functional, S, can be expanded in a Taylor series
Or the finite difference is
where it is understood that we evaluate S using the functions 
We can change notation so that 
and 
and get
The first variation of S, labeled 
, which is called the first functional derivative of S, is the linear part of
the Taylor expansion, or
and the term with second power in 
and 
is called the second variation or second functional derivative, or
and similarly for higher order variations, 
And with this notation,
What is 
?
But y(x) and z(x) do not depend on one another, so terms like 
are obviously zero.
And with terms like 
we have that 
is a function of x and is independent of y. So they go to zero as well. What
is left is
But this last part was exactly how 
was defined. So we have
So what is the variation of a composite functional? If
, then what is 
?
Remember from post 10,
But we know from calculus that,
and 
So that,
Or,
Now if,
then,
So that,
commutes with 
Then,
And keeping only the first approximation linear terms on both sides of the
equation,
commutes with 
And if
Then
So we see here that 
commutes with 
But also,
Then keeping first approximation linear terms on each side of the equation we
see
Or, 
Many text use the integral definition of a functional, 
in its development. I suppose they do this because it makes it easier to
justify taking only the linear terms inside the integral since differentials
approach zero more naturally in the process of integration.
It's easy to see that variation commutes with any number of integral signs
since a difference outside the integral is translated to a difference inside
the integral signs. Or,
So following a similar procedure of keeping only linear terms,
This means that the variation of the integration of the path integral gets
passed inside all the infinite number of integrations to taking the variation
of the exponent of the action.