Equation of motion of Nambu-Goto action with respect to $X^\mu$
I'm trying to derive the following equation of motion: $$ \partial_\alpha(\sqrt{-h}h^{\alpha\beta}\partial_\beta X^\mu)=0 $$ from the <PERSON>-Goto action: $$ \mathcal{S}{NG}=-\frac{1}{2\alpha}\int{d^2 x\sqrt{-h}} $$ where $\sqrt{-h}:= \sqrt{-det(h{\alpha\beta})}$ and $ h_{\alpha\beta} := \partial_\alpha X^\mu \partial_\beta X_\mu$.
What I have done:
We use $$\frac{\delta S_{NG}}{\delta X}=0$$
Now to find the variation of the action we need to use$$ \delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{\alpha\beta}\delta g^{\alpha\beta} $$
so $$ \delta S= -\frac{1}{2\alpha}\int{d^2 x(-\frac{1}{2}\sqrt{-h}\partial_\alpha X^\mu\partial_\beta X_\mu\delta h^{\alpha\beta})} $$
My problem is how is then $\frac{\delta S}{\delta X}$ going to give the form I mentioned in the beginning.
Edit:
I'd like to thank @Darth_Bane for his/her beautiful answer. I'll type it here with some details for the interested reader who, like me, needs to go through it step by step. 818