Consider a vector field $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$, I have that the directional derivative (or Lie derivative) can be defined as:
$L_f = \sum_{i=1}^{n}f_i \frac{d}{dx}$
which as far as I have understood is the derivative of the vector field $f$ along the $x$ direction (but I am not sure, please correct me if I am wrong).
Now, consider a derivable function $h:\mathbb{R}^{n}\rightarrow \mathbb{R}$ , I have that the operator $L_f$ acts on the function $h$ as:
$L_f h= \sum_{i=1}^{n} \frac{dh}{dx}|_x f_i(x) = [\frac{dh}{dx_i}|_x ...\frac{dh}{dx_n}|_x]\begin{bmatrix} f_1(x)\\ ... \\ f_n(x) \end{bmatrix} = \frac{dh}{dx}|_x f(x) $
but what does it mean? what does it mean that the directional derivative acts on $h$?
I have a lot of confusion on this, for example I have also that :
$L_g L_f h|_x = \frac{dL_fh|_x}{dx}g(x)$
which I don't understand. I am trying to apply these concepts to control theory, but I cannot understand how they work.
Can somebody please help me?