Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary.
What is the relation between$^1$ $T_x\:M$ and $T_x\:\partial M$ for $x\in\partial M$?
I've tried to consider the case $k=d$ and $M=\mathbb H^k$. Let $x\in\mathbb H^k$. Then we should have $T_x\:\mathbb H^k=\mathbb R^k$: Let $v\in\mathbb R^k\setminus\{0\}$ and $$\gamma(t):=x+tv\;\;\;\text{for }t\in\mathbb R.$$ Then, $$\forall t\in\mathbb R:\gamma(t)\in\mathbb H^k\Leftrightarrow x_k+tv_k\ge0.\tag1$$ We need to show that there is a nontrivial interval $I\subseteq\mathbb R$ with $0\in I$ such that $\left.\gamma\right|_I$ is a $C^1$-curve on $M$ through $x$. If $x\in(\mathbb H^k)^\circ$, then $$B_\varepsilon(x)\subseteq\mathbb H^k\tag2$$ for some $\varepsilon>0$ and hence we may choose $$I:=\left(-\frac\varepsilon{\left\|v\right\|},\frac\varepsilon{\left\|v\right\|}\right).$$ Otherwise, $x\in\partial\mathbb H^k$ and hence $$\forall t\in\mathbb R:\gamma(t)\in\mathbb H^k\Leftrightarrow tv_k\ge0\tag3$$ and we may choose $$I:=\begin{cases}[0,\varepsilon)&\text{, if }v_k\ge0\\(-\varepsilon,0]&\text{, if }v_k\le0\end{cases}$$ for some arbitrary $\varepsilon\in(0,\infty]$.
Turning to $T_x\:\partial M$ for some $x\in\partial M$: I guess the crucial difference between my proof of $T_x\:\mathbb H^k=\mathbb R^k$ above is that we now need to choose a curve which remains in $\partial\mathbb H^k$. So, if $\gamma$ is any $C^1$-curve on $\partial\mathbb H^k$, then I guess it must hold $\gamma'(0)_k=0$, since otherwise $\gamma(t)\not\in\partial\mathbb H^k$ for $|t|$ sufficiently small. So, we should have $T_x\:\partial\mathbb H^k=\partial\mathbb H^k$ or am I missing something?
EDIT
What I wrote above does hold more generally: If $(\Omega,\phi)$ is a $k$-dimensional $C^1$-chart of $M$, $x\in\Omega$, $u:=\phi(x)$ and $h\in\mathbb R^k$, then the same proof as above yields that $$v:={\rm D}\phi^{-1}(u)h\in T_x\:\Omega\tag4.$$ We just need to replace $\gamma$ by $\gamma(t):=\phi^{-1}(u+th)$ for $t\in I$ (and $\varepsilon$ needs to be chosen sufficiently small).
This shows that $${\rm D}\phi^{-1}(\phi(x))\mathbb R^k\subseteq T_x\:\Omega\subseteq T_x\:M\tag4\;\;\;\text{for all }x\in\Omega,$$ but are both subset relations in $(4)$ actually equalities?
$^1$ Say that $(I,\gamma)$ is a $C^1$-curve on $M$ through $x\in M$ if $I\subseteq\mathbb R$ is a nontrivial interval with $0\in I$ and $\gamma:I\to M$ is $C^1$-differentiable with $\gamma(0)=x$. Let $$T_x\:M:=\{\gamma'(0):\gamma\text{ is a }C^1\text{-curve on }M\text{ through }x\}$$ denote the tangent space of $M$ at $x\in M$.