in a class introducing Spacetime in General Relativity,
Given the following:
$(1)$ $f$ is a function from $d$-dimensional manifold $M$ to $R$.
$(2)$ $\gamma$ is a map from $R$ to $d$-dimensional manifold $M$
$(3)$ $V_{\gamma,p}$ (so called 'velocity') maps a smooth function $f$ to an element in $R$: $V_{\gamma,p}=(f\circ \gamma)'(\gamma^{-1}(p))$
$(4)$ $\gamma (0)=p$
the following derivation is made:
$$V_{\gamma ,p}(f)=(f \circ \gamma)'(0)=(\underbrace{(f\circ x^{-1})}_{map: R^d\to R}\circ \underbrace{(x\circ \gamma)}_{map: R\to R^d})'(0)= \underbrace{({x\circ \gamma)^i}'(0) \cdot \delta_i (f\circ x^{-1})(x(p))}_{\text {What exactly is going on here?}}$$
I don't understand the final step in this derivation. Specifically, I don't understand what exactly is meant by the notation. In another context, I would assume that it refers to a single element $i$ of the dot product of the gradient of $f\circ x^{-1}$ with the derivative vector of $x \circ \gamma$. However, does it refer to the entire dot product here? If so, the notation is weird.
So what does the final part of the derivation mean exactly?
Should the final part of the equation simply be the following?
$$\sum_{i=1}^d\left(({x\circ \gamma)^i}'(0) \cdot \delta_i \left(f\circ x^{-1}\right)\left(x(p)\right)\right)$$
i.e. was it simply written down sloppily?