b:

Because $\gamma(T)\subset\xi(U)$, there exist functions of $T$, $u(t)$ and $v(t)$ such that:

$$\forall t\in T, \gamma(t)=\xi(u(t), v(t))$$

This equation's derivative in $t$:

$$\gamma'(t_0) = u'(t_0) \xi_u(u_0, v_0) + v'(t_0) \xi_v(u_0, v_0)$$

where $\xi_u=\frac{\delta\xi}{\delta u}$ and $\xi_v=\frac{\delta\xi}{\delta v}$.