Let $M$ be a manifold, $x\in M$ and $\gamma\colon (-1,1)\to M$ be a smooth curve such that $\gamma(0)=x$.
Premise Assume that $T_x M=\{ X\colon C^\infty(M)\to \mathbb R \mid X \text{ is linear and satisfies Leibniz rule at } x\}$. Then I can define $$\dot\gamma(0)\colon f\in C^\infty(M)\mapsto \frac{\mathrm d}{\mathrm dt}\bigg|_{t=0}(f\circ \gamma)(t)\in \mathbb R$$
My question Let's assume now that $T_xM=\{ [\gamma]_x\mid \gamma\colon (-1,1)\to M,\gamma(0)=x\}$, where two curves $\gamma_1,\gamma_2\colon (-1,1)\to M$ are said to be equivalent iff there exists a chart $(U,u)$ at $x$ such that $$ \frac{\mathrm d}{\mathrm dt}\bigg|_{t=0}(u\circ \gamma_1)=\frac{\mathrm d}{\mathrm dt}\bigg|_{t=0}(u\circ \gamma_2)$$
With this definition of tangent space, is the following a good definition? $$\dot \gamma(0):=[\gamma]_{\gamma(0)}$$
Thanks in advance.