i'm currently reading An Introduction to Morse Theory by Yukio Matsumoto and on p.62 it says
A vector field itself is sort of a differential operator, since it assigns to each point a "tangent vector" which is a differential operation. Let us differentiate $f$ with respect to the gradient vector field $X_f$:
$$X_f \cdot f = \left(\sum_{i=1}^m \frac{\partial f}{\partial x_i} \frac{\partial}{\partial x_i}\right)\cdot f = \sum_{i=1}^m \left( \frac{\partial f}{\partial x_i}\right)^2 \ge 0$$
I would love to understand why $$ \left(\frac{\partial}{\partial x_i}\right)\cdot f = \left( \frac{\partial f}{\partial x_i}\right)$$
Could anyone help me on this? Thank you very much.