I wish to take the time derivative $\frac{d}{dt}$ of the expression $Lsin(\phi)$, where $\phi = \phi(t)$, but that is all that is known about $\phi$.
Using the chain rule, I obtain $\dot{\phi}Lcos(\phi)$. Is that correct? If not, what is the correct answer? Is it simply $Lsin(\dot{\phi})$?
I realize this is an elementary question, but it's been about 15 years since I took calculus and haven't used it much since then. I appreciate any help.
Nicely done! The Chain Rule is all you need to prove you first answer correct. Moreover, it's all you can use, given the information provided.
The only potential problem is this: $\phi$ was never specified to be a differentiable function, just a univariate function in $t.$ If it is safe to assume differentiability, you're fine. Otherwise, there needn't be any answer at all.