Let $X$ be a singular hypersurface in $\mathbb{C}^n$ defined as the zero locus of $f(x_1, \dots, x_n)$ and denote $Jac(f)=\langle \partial_1(f), \dots, \partial_n(f) \rangle$, where $\partial_i(f)$ is the partial derivative of $f$ with respect to $x_i$.
If we blow up $\mathbb{C}^n$ with respect to its singular locus, then on each affine chart $U_i$ of the blow up, the strict transform of $X$ is a hypersurface as well, which is defined by some $f_i$.
My question is, what is the relationship between $Jac(f)$ and $Jac(f_i)$?
(I am specifically interested in the case where the restriction of the blow up map to the exceptional divisor is smooth.)