see https://stats.stackexchange.com/questions/76815/multiple-regression-or-partial-correlation-coefficient-and-relations-between-th

sa https://stats.stackexchange.com/questions/73869/suppression-effect-in-regression-definition-and-visual-explanation-depiction/73876#73876

sa https://stats.stackexchange.com/questions/33888/x-and-y-are-not-correlated-but-x-is-significant-predictor-of-y-in-multiple-regr/34016#34016

$$ {\LARGE \beta_{x1}} $$
$$ \text{Beta:} \quad \beta_{x_1} = \frac{r_{yx_1} - r_{yx_2}r_{x_1x_2} }{1-r_{x_1x_2}^2}$$
$$ \text{Partial r:} \quad  r_{yx_1.x_2} = \frac{r_{yx_1} - r_{yx_2}r_{x_1x_2} }{\sqrt{ (1-r_{yx_2}^2)(1-r_{x_1x_2}^2) }} $$