real analysis – Tangent Space and Curves


Let $Ssubsetmathbb{R}^3$ be a regular surface and $Pin S$ such that $(U,phi)$ is a coordinate system of $S$ in $P$ where $(u_0,v_0)in U$ and $phi(u_o,v_0)=P$.

The tangent space of $S$ at $P$, $T_P{S}$, is given by $D_phi(u_0,v_0)(mathbb{R}^2)$.(I am reading Manfredo do Carmo’s book on differential geometry.)

My doubt is when proving that $T_pSsubset D_phi(u_0,v_0)(mathbb{R}^2)$.

We consider $v=alpha'(0)$ for some curve $alpha$ whose trace is in $S$, it is differentiable and $alpha(0)=P$. Then we consider $beta(t)=phi^{-1} circalpha(t)$ and so $D_phi(u_0,v_0)(beta'(0))=(phi circbeta)'(0)=alpha'(0)=v$. This prove the inclusion mentioned. My question is… why is $beta$ differentiable?