In algebraic geometry, if $X$ is an affine variety than the tangent space $T_pX$ of $X$ at its point $p$ is the affine variety defined as the set of zeros of the ideal $J$ generated by the linear parts $df$ of the polynomials $f$ belonging to $I(X)$ (the ideal of $X$).
If $I(X)$ is generated by $f_1,\dotsc,f_m$, that is $I(X)=(f_1,f_2,\dotsc,f_m)$, then $J=(df_1,\dotsc,df_m)$. It's very common the notation $J=I(T_pX)$ which suggests that $J$ is the ideal of $T_pX$. But we know that the ideal of an affine variety is radical. My question is: how do we know that $J$ is radical?