OPE in linear dilaton theory

So, from p.73 of Tong's notes you can see that an operator $O(w)$ should transform under $delta z =epsilon(z)$ by equation 4.12: $$delta O(w) = -Res[epsilon(z)T(z)O(w)]$$ meaning that knowing an operator's OPE with the stress tensor is knowing how it transforms under $delta z$. So first we must calculate the OPE $T(z)O(w,bar{w})$. The way we do OPEs is we sum over $textbf{all}$ allowed contractions between the operators (but not between normal ordered operators). The OPE you're trying to do is $$ T(z)X(w) = left(frac{1}{a'}:partial X(z)partial X(z): - Qpartial^2 X(z)right)X(w) $$ so in the first term you can either contract $X(w)$ with the first $partial X(z)$ or with the second, but not $partial X(z)$ with $partial X(z)$, so that will give you the same term (as I think you have already ghessed) therefore you can write the first term as $(2/a'):partial X(z):langle partial X(z)X(w)rangle$ where the normal ordering is now redundant. The second term only has one possible contraction and that is $-Qlanglepartial^2 X(z)X(w)rangle$.

To evaluate these contractions you need $langle X(z) X(w)rangle = frac{a'}{2}ln(z-w)$ which can be differentiated to give $$langle partial X(z)X(w)rangle = partial_zlangle X(z) X(w)rangle = partial_z frac{a'}{2}ln(z-w) = frac{a'}{2}frac{1}{z-w}$$ $$langlepartial^2 X(z)X(w)rangle = partial_z^2frac{a'}{2}ln(z-w) = -frac{a'}{2}frac{1}{(z-w)^2}.$$

The last ingredient you need to get only $(w)$ dependance on the RHS of $T(z)O(w,bar{w})$ is to Laurent expand $partial X(z)$ around $w$ by $partial X(z) = partial X(w) + (z-w)partial^2X(w)+O(z-w)^2$. Sometimes this expansion will give you more singular terms but in this case we only need it to first order.

Putting everything together we see that $$T(z)X(w) = frac{Qa'/2}{(z-w)^2}+frac{partial X(w)}{z-w}+n.s$$ meaning that $X(w)$ does not transform like a primary operator (because there is no $X(w)$ in the numerator of the first term, as you will probably see later there is a better candidate for a primary operator).

All that is left now is to calculate the residue of this OPE with $epsilon(z) = epsilon(w) + (z-w)partial epsilon(w)+... $ $$delta X(w) = -ResBig[(epsilon(w) + (z-w)partial epsilon(w)+...)left( frac{Qa'/2}{(z-w)^2}+frac{partial X(w)}{z-w}+n.s right)Big] $$ $$=-ResBig[ frac{partialepsilon(w)Qa'/2+epsilon(w)partial X(w)}{z-w} Big]$$ $$=-frac{Qa'}{2}partialepsilon(w)-epsilon(w)partial X(w)$$