Let $\sigma:V\rightarrow S$ be a regular patch. Then for all $p=\sigma(u_0,v_0)$ we define the positive normal vector $$\textbf{N}_p:=\frac{\sigma_u\times \sigma_v}{\|\sigma_u\times \sigma_v\|}(u_0,v_0)$$
For any parametrised surface $S$ we can define the gauss map $$G:S\rightarrow \Bbb{R}^3; ~p\mapsto \textbf{N}_p$$
Now suppose I take a curve on the surface $\gamma(t)=\sigma(u(t), v(t))$ in a patch such that $\gamma(0)=p$. I would like to compute $DG_p(\gamma'(0))$, i.e. the differential. The following are my thoughts about it: $$\begin{align}DG_p(\gamma'(0))&=\frac{d}{dt}\left(\textbf{N}_{\gamma'(t)}\right)\big|_{t=0} \end{align}$$Now I'm a bit unsure about how to continue. I know that $\gamma'(t)=\sigma_u\left(u(t),v(t) \right)\cdot u'(t)+\sigma_v\left(u(t),v(t)\right)\cdot v'(t)$, but how can I use this to continue?
Can someone help me?