Difficulty following chain rule segment for proof of a Euler-Lagrange energy conservation proof

Question

If you take a look at the lecture notes I'm not too sure how he's used the chain rule to rewrite the derivative in terms of $t'$ rather than epsilon.

Thanks in advance, would've typed it up but didn't want to make any mistakes transcribing and I think they're pretty readable

Screenshot from online notes

Answer 1

It seems OP's question is spurred by the fact that it is implicitly understood that $\frac{\partial}{\partial t^{\prime}}$ is an explicit (as opposed to a total) $t^{\prime}$-derivative.