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**Derivative of a function with respect to its first derivative??**

First, I will apologize for my inability to do Latex. Second, I am providing some background on my confusion. Recently, I have been learning functional derivatives and ran into the Euler-Lagrange equation. When I was reading the derivation, I had some confusion in the use of chain rule.

With my notation being, S = Integral of L( q, q', t) dt, when the partial derivatives are taken to find the change in functional S, and we end up with dS/de= Integral of ( dL/dq * dq/de + dL/dq' * dq'/de ) dt.

Where the d's are replaced by the appropriate partial or functional derivative. Now, my question about the chain rule is, is it possible to turn dL /dq' into dL/dq * dq/dq'?

Does dq/dq' exist? And if it does, is it a total derivative, partial derivative, or functional derivative?

Sorry if this seems a little bit confusing or if I am formulating my question incorrectly.