Bound for the derivative of a $C^1$ curve

Question

I have a $C^1$ curve $\gamma:(a,b)\rightarrow \mathbf{R}^n$, such that $\gamma(a,b)$ is contained in some compact set. How do I get a bound for $\gamma^{\prime}$?

My Attempt: for any $t\in(a,b),||\gamma^{\prime}(t)||=||D\gamma_t (1)||\leq ||D\gamma_t ||$. But still can't get rid of $t$. Is it always even bounded?

Answer 1

$\gamma (t)=\sqrt t$ from $(0,1) \to \mathbb R$ shows that $\gamma'$ need not be bounded. If you want an example in which the domain is a closed interval consider $x \to x^{3/2} \sin (\frac 1 x)$ on $[0,1]$. (Assign the value $0$ at $x=0$).

Answer 2

You can't prove it, since it is false. Take, for instance,$$\begin{array}{rccc}\gamma\colon&(-1,1)&\longrightarrow&\mathbb{R}^2\\&t&\mapsto&\left(t,\sqrt{1-t^2}\right).\end{array}$$