In our lecture, we have defined the representation of a Lie algebra $\mathfrak{g}$ on the space of linear maps $L(V,W)$ between two representations $V$ and $W$ of $\mathfrak{g}$ to be of the following form:
$$(X \cdot \phi)(v) := X \cdot (\phi(v)) - \phi(X \cdot v) $$
It is explained that this is the differentiated version of the group action of a group $G$ on the set of all functions $\mathcal{F}(X,Y)$ of two sets $X$ and $Y$ defined by
$$ (g\cdot \phi)(x) := g \cdot(\phi(g^{-1}\cdot x)) $$
I have two questions here:
Why do we differentiate it?
And more importantly: How does one differentiate it? With respect to what? I have been trying to calculate it but failed.
By the way, this is not a homework question, it is just not explained in the notes.