Dependance of variables in variational calculus

Question

I stumbled across the following functional for which a stationary path should be found: $$ L[y]=\int\limits_a^b x[y‘(x)]^4+ ... \,\mathrm{d}x$$ Where the dots indicate the other terms that were included in the original problem but dont cause any confusion. The first question that arises is how to make any sense of this expression since both $y$ and $y‘$ are functions of $x$ but then $x$ itself happens to depend on $y‘$ which then again is a function of itself (?). I hope someone can elude me, or at least confirm that it really does not. The next step would then be to differentiate this partially with respect to $y$ and $y‘$ in order to be able to use the Euler-Lagrange equation. If the first term that causes the problem does not reduce to $0$ then how could one possibly find a solution to a differential equation where the variable depends on the function itself?

Answer 1

I think that you just kind of misread. Isn't it $x \cdot [y'(x)]^4$, i.e. $x$ times $y'^4$?