Tangent plane and differential Let $M\subset \mathbb{R}^m$ a submanifold of dimension $d$ and let $g:V_p\longrightarrow \mathbb{R}^{m-d}$ a submersion defined in a neighborhood $V_p$ of $p\in M$ such $V_p\cap M = g^{-1}g\lbrace p \rbrace$.

I would like to prove that $\mathrm{T}_pM \subset \ker \mathrm{D}_pg$. (And to conclude to the egality using a dimension argument).

Let $x\in\mathrm{T}_pM:=\lbrace \gamma '(0)\in\mathbb{R}^m \mid \gamma : (-\varepsilon ,\varepsilon )\longrightarrow M \text{ such that } \gamma (0) =p \rbrace$. The condition $V_p\cap M = g^{-1}g\lbrace p \rbrace$ implies that $g$ is constant under $V_p\cap M$. We have
$$\mathrm{D}_pg(x)=\mathrm{D}_{\gamma (0)}g(\gamma '(0))=\frac{\mathrm{d}}{\mathrm{dt}}g(\gamma (t))$$
And I would like to conclude that $\frac{\mathrm{d}}{\mathrm{dt}}g(\gamma (t))=0$ because $g$ is constant. And so $x\in\ker \mathrm{D}_pg$. My problem is that we don't know if $\gamma (t)$ is in $V_p\cap M$ so I can't conclude that $\frac{\mathrm{d}}{\mathrm{dt}}g(\gamma (t))$ is equal to $0$.
Can I have some help please ? (Sorry if my english is approximate).

As suggested by Le Van Tu, we can take $\epsilon$ to be arbitrarily small.
(Essentially this is because the tangent space consists of equivalence classes of paths or to use fancier language of germs) 
Since $\gamma$ is smooth, it is in particular continuous, hence (since $V_p \cap M$ is open in $M$) we can assume that $\gamma(t) \in V_p \cap M$ for $\epsilon$ small enough.
The rest of you reasoning is correct.
