In my book the author makes the remark: If $X,Y$ are smooth vector fields, and $\nabla$ is a connection, then $\nabla_X Y(p)$ depends on the Value of $X(p)$ and the value of $Y$ along a curve, tangent to $X(p)$.
When I got it right, then we can consider a curve $c:I\rightarrow M$ with $c(0)=p$ and $c'(0)=X_p$.
I was wondering, why this is true. When I consider a coordinate representation around the point p, i.e. $X=\sum x^i\cdot \partial_i$ and $Y=\sum y^i \cdot \partial_i$, then we can calculate that $\nabla_X Y(p)$ depends on:
$x^i(p)$, $y^i(p)$ and $X_p(y^i)$.
This again is only depending on $X_p$ and the values of $y^i=Y(x^i)$ in a arbitrary small neighborhood of $p$. But I cannot see any curve ...
I hope you can help me!