If this is your first time doing differential geometry you should do the calculation that ayanta is referring to in their comment. Let $X \colon C^\infty(M) \to \mathbb{R}$ be a linear map satisfying the Leibniz rule $$ X(fg) = X(f) g(x) + f(x) X(g) $$ for all $f, g, \in C^\infty(M)$. Show that if $\psi = (\psi^1, \dots, \psi^n)$ are co-ordinates in any neighbourhood of $x$ there are real numbers $X^1, \dots, X^n$ such that for all $f \in C^\infty(M)$ we have $$ X(f) = \sum_{i=1}^n X^i \frac{\partial (f \circ \psi^{-1})}{\partial \psi^i} (\psi(x)) . $$

Also a good exercise to show that for the same $X$ there is a smooth function $\gamma \colon (-\epsilon, \epsilon) \to M$ such that $\gamma(0) = x$ and for any $f \in C^\infty(M)$ we have $$ X(f) = \frac{ d (f \circ \gamma) }{ dt}(0). $$ This connects you to tangent vectors thought of as equivalence classes of curves as Davidac897 discusses.

If this is your first time doing differential geometry you should do the calculation that ayanta is referring to in their comment. Let $X \colon C^\infty(M) \to \mathbb{R}$ be a linear map satisfying the Leibniz rule $$ X(fg) = X(f) g(x) + f(x) X(g) $$ for all $f, g, \in C^\infty(M)$. Show that if $\psi = (\psi^1, \dots, \psi^n)$ are co-ordinates in any neighbourhood of $x$ then there are real numbers $X^1, \dots, X^n$ such that for all $f \in C^\infty(M)$ we have $$ X(f) = \sum_{i=1}^n X^i \frac{\partial (f \circ \psi^{-1})}{\partial \psi^i} (\psi(x)) . $$

Also a good exercise to show that for the same $X$ there is a smooth function $\gamma \colon (-\epsilon, \epsilon) \to M$ such that $\gamma(0) = x$ and for any $f \in C^\infty(M)$ we have $$ X(f) = \frac{ d (f \circ \gamma) }{ dt}(0). $$ This connects you to tangent vectors thought of as equivalence classes of curves as Davidac897 discusses.