Is it true if $X$ is nonsingular affine variety of dimension $m$ inside $\mathbb{A}^n$ then $TX$, the tangent bundle of $X$ is also nonsingular variety but with dimension $2m$? I have proved it is an algebraic set of dimension $2m$ but I do not know how to prove it is irreducible and it is nonsingular. I tried to compute the rank of Jacobian but did not find the rank.
Edit : idea of my work so far is we have $TX=Z(f_1,\dots,f_r, df_1,\dots, df_r)$ where $X=Z(f_1,\dots,f_r)$. When I computed the Jacobian matrix for $TX$ at some $(p,v) \in X$, I will get lower triangular blocks of matrix, but I was not able to find its rank. The matrix will be $\begin{pmatrix} A & 0\\ B & C \end{pmatrix}$ with $A$ consists of first derivatives of $f_1,\dots,f_r$ and $B$ consists of second derivatives of same polynomials