How do I obtain a correlation between two variables?

Question

Can I differentiate a quadratic regression model formula with the form
$$Y_i=beta_0+beta_1 X_{1i} +beta_2 X_{1i}^2+epsilon_i$$
(added to this formula would be other control variables e.g dummy variables) to obtain the correlation between X and Y holding everything else constant? Also, if I were to add a female dummy variable, how would the correlation of X and Y differ between men and women? Or would the correlation be the same?

Tomcat · Accepted Answer

For the above equation is differentiable, and hence the partial derivatives will exist:
$frac{partial Y}{partial X_2}= 2beta_2X_2$.
Adding a dummy variable should not affect the above derivative for all practical purposes - the function will be differentiable a.e.
(note that you can't differentiate with respect to a dummy variable though)
Edit:
After the question was modified, the new expression of the partial derivative would be:
$frac{partial Y}{partial X_1}=beta_1+ 2beta_2X_1$.

How do I obtain a correlation between two variables?

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