Geometry has recently been giving me a slight headache, so I will write about it like I usually do in this situation. I will first write about the abstract formulation of tangent vectors.

Let $M$ be a smooth manifold, and let $p\in M$. A derivation at $p$ is a function $v:C^\infty\left(M\right)\to\mathbb R$ such that for all $f,g\in C^\infty\left(M\right)$, it is the case that
$$v\left(fg\right)=f\left(p\right)vg+g\left(p\right)vf.$$
The tangent space to $M$ at $p$, denoted $T_pM$, is the set of all derivations at $p$. Strangely, an element of $T_pM$ is called a tangent vector at $p$.

Define the geometric tangent space to $\mathbb R^n$ at $a$, denoted $\mathbb R_a^n$, to be a sort of copy of $\mathbb R^n$ but with its origin at $a\in\mathbb R^n$. Formally,
$$\mathbb R^n_a=\left\{a\right\}\times\mathbb R^n.$$
This is so that $\mathbb R^n_a\cap\mathbb R^n_b=\varnothing$ whenever $a\neq b$. For an element $\left(a,v\right)\in\mathbb R^n_a$, write $v_a$ or $\left.v\right|_a$ instead, and call it a geometric tangent vector at $a$.

Let $f\in C^\infty\left(\mathbb R^n\right)$, and recall that the derivative of $f$ in the direction of some $v\in\mathbb R^n$ at some $a\in\mathbb R^n$, denoted $\left.D_v\right|_af$, satisfies
$$\left.D_v\right|_af=\left.\frac d{dt}\right|_{t=0}f\left(a+tv\right).$$
It goes without saying (from calculus) that $\left.D_v\right|_a$ is linear and is a derivation at $a$.

Surprisingly, $\mathbb R^n_a$ and $T_a\mathbb R^n$ are isomorphic (with the map $v_a\mapsto\left.D_v\right|_a$).

Let $M$ and $N$ be smooth manifolds, and let $F:M\to N$ be a smooth map. For each $p\in M$, define a map
$$dF_p:T_pM\to T_{F\left(p\right)}N$$
and call it the differential of $F$ at $p$. Let $v\in T_pM$, let $f\in C^\infty\left(N\right)$, and define
$$dF_p\left(v\right)\left(f\right)=v\left(f\circ F\right).$$
We are now almost ready to talk about differentiating maps between manifolds.