Representation of a Lie algebra over the space of smooth functions.

Question

Suppose that $\mathfrak{g}$ is the Lie algebra of a compact and connected linear Lie group. Now let's consider the mapping $$\rho : \mathfrak{g} \longrightarrow \mathfrak{gl}({\mathcal{C}^{\infty}(G, \mathbb{C})} )$$ defined by $$\rho(X)f(x) = \frac{d}{dt}\Bigr|_{t=0}f(x\exp(tX))$$ for all $$X\in\mathfrak{g},\,\, f\in\mathcal{C}^{\infty}(G, \mathbb{C}) \,\,\,\, \mbox{and} \,\,\,\, x\in G$$ In the book, "Unitary Representations and Harmonic Analysis" by Mitsuo Sugiura, page 75, it is stated without proof, that the map $\rho$ is a Lie algebra representation. Can anyone give me any suggestions to prove that the $\rho$ map is linear?