Directional Derivatives on a Manifold. Let $M$ be a manifold (if its easier then say a Riemannian Manifold).
My aim is to understand directional derivatives of some function $\Phi: M\to \mathbb{R}$.
$\underline{\textit{My Understanding So Far :}}$

*

*For a manifold $M$ the tangent space at a point $p$ (denoted $T_p M$) can be seen as the set of equivalence classes of curves on $M$ passing through $p$. More specifically : https://en.wikipedia.org/wiki/Tangent_space#Definition_via_tangent_curves .


*A Riemannian Manifold is a manifold $M$ equipped with an inner product on $T_p M \times T_p M$ (which will let us talk about angles and lengths of curves on $M$).


*The classical directional derivative (in direction $v$ and at point $p$) of a function acting on $\mathbb{R}^d$ tells you how much the function changes at point $p$ and in direction $v$.
$\underline{\textit{Help : }}$
Im finding it hard to go from the above to the definition of directional derivatives given here https://en.wikipedia.org/wiki/Tangent_space#Tangent_vectors_as_directional_derivatives . Any help ? :)
 A: I think it's helpful to think about how we compute the directional derivative in $\mathbb{R}^n$, this is done as $$D_vf(x_0)=\lim_{h\to 0} \frac{f(x_0+hv)-f(x_0)}{h}$$
If we have some differentiable curve $\gamma: I\to \mathbb{R}^n$ such that $\gamma(t_0)=x_0$ then $\frac{d}{dt}f(\gamma(t))$ at $t_0$ is given by the directional derivative of $f$ in the direction of $\dot{\gamma}=v$, i.e. $$\frac{d}{dt}f(\gamma(t))=\nabla{f}\cdot v=v[f]_{x_0}.$$ This implies that we can think of $\dot{\gamma}$ as the directional derivative operator satisfying $$\frac{d}{dt} f(\gamma(t)):=\dot{\gamma}[f]_{x_0}$$ Since there are many such curves that give the same result we can take an equivalence class of them. In the end we don't need the Riemannian structure on $M$ in order to talk about these vectors, we only need a differentiable structure on $M$ in order to get consistent answers for our derivatives. When going to a manifold, we'll have the map $\gamma: I\to M$ the map $f: M\to \mathbb{R}$ meaning that $f\circ\gamma: I\to \mathbb{R}$ where we can do normal calculus and get coordinate independent results.
