For reference, I shall use the notation used in the wikipedia article for the Beltrami identity in the application section (http://en.wikipedia.org/wiki/Beltrami_identity)
In the article, the example of the Brachistochrone problem is used. It is stated that the function inside the integral does not depend explicitly on x, so the Beltrami identity applies. The Beltrami identity is stated to apply where dL/dx = 0.
What I don't understand is why x not explicitly appearing in L is sufficient to state that the Beltrami identity applies. dy/dx appears in L, so why can't x appear within dy/dx? wouldn't that then make it so that dL/dx is not necessarily 0?