Roadmap to Euler Lagrange Equation

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[barryhughes]

Hi again Ross. The part of this that confuses me in this example is why stop at the first derivative, that is the gradient change. I mean the second derivative must change as a consequence of a change in the first derivative. I mean we must consider all the changes, both f[x] say, [ height[ and also all the changes due to a change in the gradient. Why stop at the first derivative?
As always Ross, thanks for your great videos and explanations.