Proving that e^{ikx} is primary with weight (h=\hbar = \alpha k^2/4)

[LCSphysicist]

Homework Statement

I can't understand the line of reasoning used by David Tong (on its lectures of CFT).

Relevant Equations

.'

Where

$:$ really means normal ordered, in the sense that $:A(w)B(z): = \lim_{w \to z} \left ( A(w)B(z) - \langle A(w)B(z) \rangle \right )$

$\partial X(z) = \frac{\partial X(z)}{\partial z}$

How do we go form the first line to the second one?? I am not understanding it!

it seems to me that we start with
$$\partial X(z) : X(w)^n : = \partial X(z) : X(w)^{n-1} X(w) :$$
Then, for some reason

$$\partial X(z) : X(w)^{n-1} X(w) : \rightarrow n X(w)^{n-1} :\partial X(z) X(w): $$

Since

$$: \partial X(z) X(w) = \frac{-\alpha'}{2 (z-w)} $$

We got the answer, but how?

 

[Demystifier]

You must use Wick theorem, which is the same as in ordinary QFT.