# Interpreting marginal effects for multivariate linear regression; what to make of controls?

Given a linear model like: $$y = \alpha + \beta_{1}X1 + \beta_{2}X2 + \beta_{3}X3$$

And if I want to interpret the marginal effects of $$\beta_{1}$$, then I am wondering about its interpretation.

I would calculate its slope along X1 values of interest by doing something like:

Marginal Effect: $$\mu_{i} = \alpha + \beta_{1} * X_{i}1$$

The uncertainty that I have is; I calculated the value of $$\beta_{1}$$ conditional on the $$\beta_{2}$$ and $$\beta_{3}$$; so the parameters were updated together. The marginal effect, however, does not take into account the covariates explicitly, although like I said, $$\beta_{1}$$ was optimized given $$\beta_{2}$$ and $$\beta_{3}$$.

So, I am wondering if I would still say something like:

A one-unit increase in X1 is associated with a n change in $$\mu$$, on average, after controlling for $$\beta_{2}$$ and $$\beta_{3}$$. Or, should no claim be made to the other variables?