(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I'm working on a quick problem regarding a presentation that I'm giving, but I've come across an issue that I can't seem to resolve. Namely

[tex] \displaystyle \left. \frac{d}{dt} \right|_{t=0} f(\phi^p (t+t_0) ) = \left( \phi^p \right) ^\prime (t_0) f [/tex]

Does anybody see how this is true?

3. The attempt at a solution

[tex] \displaystyle \left. \frac{d}{dt} \right|_{t=0} f(\phi^p (t+t_0) ) = f^\prime(\phi^p(t_0)) \left(\phi^p \right)^\prime (t_0) [/tex]

All we know about f is that it is a smooth function and [itex] t_0 [/itex] was arbitrarily chosen, so I'm not seeing where we make the jump. (Note: [itex] \phi^p(t) [/itex] is a dynamical system flow on a smooth manifold, but I don't see how that should help)

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# Homework Help: Weird Issue with the Chain Rule

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