Derivative of a Functional w.r.t. a compound function

Question

I have a functional of the form

$ F(x, f(x), f(g(x)))$

e.g.

$F=e^x f(x)-f(e^x)$

I want to compute the derivative with respect to $f(.)$ but I am not sure how to. My intuition is that the function $g(x)$ does not matter for the derivative and therefore:

$\frac{\delta F}{\delta f}=e^x-1$

But I want to be sure, thanks in advance.

Answer 1

Generally, no, $g(x)$ does matter. Take $g(x)=x$ for example and you will see that something does matter about $g$.

To be more specific, the derivative with respect to $f$ means:

As $f$ changes, how much does $F$ change? Then take the ratio of these two (with a limit)

Particularly, assuming $f(x)$ is an actual function, then as $f$ changes, $x$ changes.

As $x$ changes, $g$ changes.

And as $x$, $f$, and $g$ change, $F$ changes.

The exact relationship is given by

$$\frac{dF}{dy}=\frac{\partial F}{\partial x}\frac{dx}{df}+\frac{\partial F}{\partial f}\frac{df}{dy}+\frac{\partial F}{\partial f(g)}\frac{df(g)}{dy}$$

Which is the multivariable chain rule. One might wish to use the relationship:

$$f(g(x))=f(g(y^{-1}(y(x))))$$

Assuming $y^{-1}$ exists. Presumably, it must exist, as there must be one unique $x$ for each $y$ on the interval we are looking at, or else the line:

As $y$ changes, $x$ changes

fails to be unique, being that there are multiple values of $x$ for each $y$, each changing at different rates.