Let $\Omega\subset{\mathbb R}^m$ be an open set and $h\colon \Omega\to {\mathbb R}^{k-m}$ be a smooth map. Prove that the tangent space of the graph of $h$ at a point $(x,h(x))$ is the graph of the differential $df(x)\colon{\mathbb R}^m\to{\mathbb R}^{k-m}$: $$M=\{(x,h(x))\mid x\in\Omega)\}, T_{(x,h(x))}M=\{(\xi,dh(x)\xi)\mid \xi\in{\mathbb R}^m)\}.$$
Can anyone give some hints please.
Let $x \in \Omega$. Let $r > 0$ such that $B(x;r) \subset \Omega$. Consider the parametrization $\phi : B(0;r) \to M$ defined by the equation $$\phi(v) = (x + v, h(x+v)).$$ It maps $0$ to $(x,h(x))$, so $T_{(x,h(x))}M$ is the image of $d\phi(0)$. Now show that for all $\xi \in \Bbb R^m$, $$d\phi(0)(\xi) =(\xi, dh(x)(\xi)).$$