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?