According to page 7 of the PDF document
$$ \frac{\delta}{\delta f} \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} dx $$
I would like help proving this statement.
Although I can show that
$$ \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} f dx $$
My attempts at "constructing" the functional derivative of this expression isn't dropping the term $f$. I'm not even sure this is the right way to go about solving the problem.
Well, probably this is a case of confusing notation. Especially I am not sure where you want to have your exponents.
In my understanding, both equations you stated are just wrong. However, the first one may carry some truth: Consider the functional $J(f) = \int (\frac{d^2 f}{dx^2})^2 dx$. Then the functional derivative of $J$ (or first variation) of $J$ is (by integration by parts) $$ \delta J(f)(h) = \int 2\frac{d^2f}{dx^2}\frac{d^2h}{dx^2}dx = \int 2\frac{d^4 f}{dx^4} h dx. $$ Hence, one may say (if put in a proper framework of function spaces) that the derivative of $J$ is $$ J'(f) = 2\frac{d^4 f}{dx^4}. $$
To conclude: The pdf document you linked is very sloppy with the notation and probably you may consult a book one the calculus of variations to get more information here.