Derivation of Snell's law

Fermat's principle says that the path taken between two points by a ray of light is the path that can be traversed in the least time.

How long does it take for light to pass from $A_1$ to $A_2$? If $c_1, c_2$ are the speeds of light passing through media $1,2$ respectively, then the time needed will be $$t(x)=\dfrac{\sqrt{h_1^2+x^2}}{c_1}+\dfrac{\sqrt{h_2^2+(a-x)^2}}{c_2}.$$ Then at the critical point, $$t'(x)=\dfrac{1}{c_1}\dfrac{x}{\sqrt{h_1^2+x^2}}-\dfrac{1}{c_2}\dfrac{a-x}{\sqrt{h_2^2+(a-x)^2}}=0.$$ Now comes the surprising moment: $$\dfrac{x}{\sqrt{h_1^2+x^2}}=\sin \alpha_1,\quad \dfrac{x}{\sqrt{h_1^2+x^2}}=\sin \alpha_2.$$ This actually gives us $$\dfrac{\sin \alpha_1}{c_1}=\dfrac{\sin \alpha_2}{c_2}$$ or $$\dfrac{\sin \alpha_1}{\sin \alpha_2}=\dfrac{c_1}{c_2},$$ which is Snell's law. In words, it says that when the sines of the angles in the different media are in the same proportion as the propagation velocities, the time of travel from $A_1$ to $A_2$ is minimised.

Reference:
Mathematical Analysis I by Vladimir A. Zorich