I am new to studying differential forms, and had questions while going over some readings.
Let $M$ be a $C^{k}$ manifold of dimension $n$, $\gamma:[0,1]\rightarrow M$ a smooth curve, and $\gamma_s$ a smooth variation of $\gamma$ that agrees with $\gamma$ at $0$ and $1$. We then have that $ \frac{d}{ds}\Big|_{s=0} \gamma_s$ is a globally defined vector field on $\gamma_s(M)$, where the derivative is understood to be an element of the tangent bundle.
Let $(x^1, \ldots, x^n)$ be coordinates of some chart defined on an open subset $\gamma(M) \subset U$ of $M$. I do not understand why $$\sum_{i=1}^n \frac{d}{ds} \Big|_{s=0}x^i(\gamma_s) = \sum_{i=1}^n dx^i (\frac{d}{ds}\Big|_{s=0}\gamma_s)$$ ($dx^i$ is a chart-induced one-form). It is clear to me that the right hand side of the equality is a sum of the local coordinates of the global vector field, but I don't know how to formally manipulate the one-form.
Further, if $p^i = dx^i$, why is the vector field $\sum dx^i(\frac{d}{ds} \Big|_{s=0} \gamma_s) \frac{\partial}{\partial p^i}$ invariant under coordinate transformations? I actually do not properly understand what it means to differentiate with respect to a one-form. I thought about expressing $p^i$ in terms of the holonomic basis induced by the chart to reduce the expression to the global vector field above, but I don't know if this will work.
Please try to make your answer as explicit as possible.
Important edit: After a lot of confusion and careful thought, I have determined there is a possibility the material I was reading meant to imply $$dx^i (\gamma'(0)) = \frac{d}{dt}(x^i \circ \gamma)(t) \Big|_{t=0}.$$
Would this make things consistent? If yes, is this standard notation?