In the standard probit regression we have that $ Pr(y>0| X) = \Phi(\frac{X\beta}{\sigma})$.
With $\Phi$ Cumulative Density Function of the Normal Distribution.
The marginal effect of the covariate $ x_j$ is:
$\phi(\frac{X\beta}{\sigma}) \beta_j$
Where $X$ is the nxk vector of covariates, and $\beta$ is the kx1 vector of coefficients and $\phi$ Probability Density Function of the Normal Distribution.
Now I want to calculate the derivative of this marginal effect with respect to the parameter $\beta_j$, namely
$\frac{\partial \phi(\frac{X\beta}{\sigma}) \beta_j}{\partial \beta_j }$
Using the Chain rule I have:
$\frac{\partial \phi(\frac{X\beta}{\sigma}) \beta_j}{\partial \beta_j } = \phi(\frac{X\beta}{\sigma}) +\frac{\partial \phi(\frac{X\beta}{\sigma})}{\partial \beta_j } \beta_j $
How do I compute $\frac{\partial \phi(\frac{X\beta}{\sigma})}{\partial \beta_j } $ ?