I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function.
In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is the Fourier transform of f, is there a simple expression for $\dfrac{\partial g}{\partial f} $? Thanks in advance for any insight or reference!
You have $g=F(f)$. The Fourier transform is a continuous linear map in many good spaces (Schwartz space, its dual, or $L^2$). The derivative of a linear map is the map itself, and so a continuous linear map is continuously differentiable (and the second derivative vanishes). That is, $\partial g/\partial f=F$. (One usually denotes this by $DF(f)=F$ or something similar.)
Let me elaborate on what this means. Consider the problem in $L^2$, for example. Recall that $F:L^2\to L^2$ is continuous. Fix a point $f\in L^2$. The derivative of $F$ at $f$ is a linear map $L:L^2\to L^2$ that satisfies $$ F(f+h)-F(h)=L(h)+o(\|h\|). $$ Such a map $L$ is unique if it exists (it is a nice little exercise to see why). If we choose $L=F$, this is true, and the error term is actually zero. Therefore a linear map $L$ exists, and by uniqueness the derivative is indeed $F$.
To learn more about these ideas, try searching for "differential calculus in Banach spaces" or something similar. Differentiating a function between Banach spaces is quite similar in spirit to differentiating one between Euclidean spaces; if you know how to pass from linear algebra to functional analysis, you are ready to study it.