About vector fields in manifolds

Question

maybe this question is to basic, but I am a litle bit confused with the definition of vector field in a manifold. Following this cuestion Vector field on manifold, I know that I can see a vector field as a derivative, and it can be expressed in the basis of the tangent space, i.e. if $X$ is a vector field, then if we restrict to a neighborhood $U$ of the manifold \begin{equation} X=\sum _{i=1}^{n} f_i \frac{\partial}{\partial x_i} \end{equation} So, if I take a fonction $f$ smooth enough, we have that \begin{equation} X(f)=\sum _{i=1}^{n} f_i \frac{\partial}{\partial x_i}(f) \end{equation} Is this last statement correct?