Let $M$ be a smooth $n$-manifold, let $f(t,y):(-\epsilon,\epsilon)\times M\to\mathbb R$, and let $\Phi(t,x):(-\epsilon,\epsilon)\times M\to M$ be defined so that given each $t$, $\Phi(t,x)$ is a diffeomorphism. Then we define a vector field $X_t$ so that $X_t(\Phi(t,x))=\frac{\partial}{\partial t}\Phi(t,x)$. I was wondering if we have a chain rule for $\frac{\partial}{\partial t}f(t,\Phi(t,x))$ using the information above. Thank you.
This question is easy to tackle in the setting of Euclidean spaces, but, with $x$ and $y$ both ranging over an abstract manifold, I'm stuck: $$\begin{align} \frac{\partial}{\partial t}f(t,\Phi(t,x))&=\frac{\partial f}{\partial t}\frac{\partial t}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}\\ &=\frac{\partial f}{\partial t}(t,\Phi(t,x))+\frac{\partial f}{\partial y}(t,\Phi(t,x))\frac{\partial \Phi}{\partial t}(t,x) \end{align}$$ How do we interpret $\frac{\partial f}{\partial y}$? I looked through my DG books but didn't find anything. Do we have to impose a Riemannian metric on $M$ to discuss this question? Could someone please shed light on this topic? Thank you.