If the length of a curve equals to the interval, than the derivative is $1$

Question

Let $X(s)$ be a curve for $0 \le s \le L$, where $L_{[0,s_0]}(X) = s_0$ for every $0 \le s_0 \le L$.

Prove that if $X\in C^1([0,1])$ then $|X'(s)| = 1$ for every $0 \le s \le L$.

Any hints on that?

I know the formula - $L_{[0,s_0]}(X) =\int^{s_0}_0|X'(s)|ds$ and yet I'm not sure where to start... any ideas?

Answer 1

You have that $$ \int_0^{s_0} |X'(s)|\,ds = s_0. $$ Now apply $\frac{d}{ds_0}$ to both sides and use the Fundamental theorem of calculus.