Could someone help me to solve this limit?
Let $X:\Omega \subset \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$ imersion and ${\LARGE \chi }_{t}:\Omega \subset \mathbb{R}^{2}\rightarrow \mathbb{R}^{3}$, $t\in \left( -\epsilon ,\epsilon \right) $ a variation of $X$ in the form ${\LARGE \chi }\left( p,t\right) =X\left( p\right) +tf\left( p\right) N\left( p\right) $ , where $N$ is normal of Gauss and $f:\Omega\rightarrow\mathbb{R}^3$ function differentiable.
Show that $$ A^{\prime }\left( 0\right) =\int_{\Omega }\left. \frac{d}{dt}\right\vert _{t=0}{Jac}({\LARGE \chi }_{t})(p)dA.$$
$A^{\prime }\left( 0\right) =\underset{t\rightarrow 0}{\lim }\frac{1}{t}% \left( A\left( t\right) -A\left( 0\right) \right) =\underset{t\rightarrow 0}{% \lim }\frac{1}{t}\left( \int_{\Omega }\sqrt{E_{t}G_{t}-F_{t}^{2}}% dudv-\int_{\Omega }\sqrt{EG-F^{2}}dudv\right) $
$=\underset{t\rightarrow 0}{\lim }\frac{1}{t}\int_{\Omega }\sqrt{% E_{t}G_{t}-F_{t}^{2}}-\sqrt{EG-F^{2}}dudv=?$