Show that $K=-\frac{\Delta \log \lambda }{\lambda ^{2}}$

Question

Let $X:\Omega \subset %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{2}\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ conformal imersion, $K$ tha a gaussian curvature of an surface $% S=X\left( \Omega \right) $ in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$.

Is there an easier way (that does not use the Christoffel symbols) to show the equality below?

$K=-\frac{\Delta \log \lambda }{\lambda ^{2}}$, where $\lambda ^{2}\left( p\right) =\left\Vert X_{u}\right\Vert ^{2}=\left\Vert X_{v}\right\Vert ^{2}$.

Everywhere I look I just think using the Christoffel symbols.