Let $f\in C^{k}(M,\mathbb R)$ with $M$ is a $m$-Manifold and $d_xf:T_xM\to\mathbb R$ is the surjectiv differential. Let $m\lt l$ and $L:\mathbb R^l\to\mathbb R^{m-1}$ be a linear map and $G:M\to\mathbb R\times\mathbb R^{m-1}$ with $G(a)=(f(a),L(a))$. Furthermore is $A=ker(d_xf)\subset\mathbb R^l$ and $L$ is regular on A. Then $d_xG(u)=(d_xf(u),L(u))$.
Question: How do I get $d_xG$?