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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.

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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.