Power analysis for linear model test I am building a linear model to analyze experimental data with the following shape:
$$
y \sim b_0 + b_1T + b_2X + N(0, \sigma^2)
$$
Where $T$ is a binary variable indicating the treatment and $X$ is a covariate.
The hypothesis test that I am going to run to check if the treatment has any significant effect is:
$$
H_0: T = 0
$$
vs
$$
H_1: T \neq 0
$$
I want to analytically obtain expressions of the power of the test, something like this but for the linear model I am writing. Can anyone share the derivation of the power of the test?
Thanks in advance!

In Regression Methods in Biostatistics, Second edition (Springer, 2012), Vittinghoff et al. provide formulas that can be adapted to your terminology.
First, the variance of the linear regression coefficient estimate $\hat \beta_T$ for predictor $T$ (treatment) in your two-predictor model is (based on their equation 4.4):
$$\text{Var}(\hat \beta_T ) = \frac{\sigma^2}{(n-1)\sigma^2_T(1-r^2_{TX})}.$$
Note that this depends on the correlation between $X$ and $T$, $r_{TX}$. More generally in a multiple-predictor model, you can replace $1/(1-r^2_{TX})$ with the variance inflation factor for $T$. A randomized trial will ideally minimize that correlation.
With a binary $T$, $\sigma^2_T=f_T(1-f_T)$, where $f_T$ is the fraction of cases in one of the two treatment categories.
The power $\gamma$ to distinguish a particular true coefficient value $\beta^a_T>0$ from 0 at a type-I error level of $\alpha$ is given by a large-sample normal approximation (their equation 4.14):
$$\gamma \approx \Phi \left[\beta^a_T/\text{SE}(\hat \beta_T) - z_{1-\alpha/2}\right],$$
where $\Phi$ is the standard normal cumulative distribution such that $\Phi(z_{1-\alpha/2})=1-\alpha/2$. Apply the inverse transformation $\Phi^{-1}$ to both sides of the power equation; $\Phi^{-1}(\gamma)=z_\gamma$. Under the large-sample approximation, replace $(n-1)$ in the variance estimate with $n$, take the square root to get the standard error $\text{SE}$, and solve for $n$ (based on their equation 4.15):
$$n = \frac{(z_{1-\alpha/2} + z_\gamma)^2\sigma^2}{(\beta^a_T\sigma_T)^2(1-r^2_{TX})}.$$
