I can't understand why textbook says that directional derivative cannot be defined for general manifold, and a separate affine connection is needed.
Assume there are two vector fields $X$ and $Y$. In particular, you have a tangent vector at point $p$ called $X_p$. To find the derivative of $Y$ at $p$ in the direction of $X_p$, why can't you just operate $X_p = a^i\partial_i$ on $Y_p = b^j \partial_j$ component-wise?
By component-wise, I meant differentiate the component smooth function $b^j$ with respect to $X_p$. It seems that a smooth real valued function can still be differentiated by a vector field without connection?