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The second expression in your derivative (the derivative of the arcsin term) is incorrect as you need to use the chain rule. $$\frac{d}{dx}\sin^{-1}(\sqrt{x})=\frac{d}{d(\sqrt{x})}\sin^{-1}(\sqrt{x})\frac{d}{dx}(\sqrt{x})=\frac{1}{\sqrt{1-x}}\frac{1}{2\sqrt{x}}$$ So the differential should be ...


2

The ramification divisor $R$ is supported at those points $x\in C$ such that $f^{-1}(f(x))$ consists of strictly less than $|G|$ points. But $f^{-1}(f(x))\subset C$ is the orbit of $x$ (as $C\to C/G$ is a geometric quotient). As such, it has cardinality $|G|/|G_x|$. Thus, $$|f^{-1}(f(x))|<|G|\iff G_x\neq 1\iff \textrm{the action is not free at }x.$$ So ...


1

Write the equation for the line through the endpoints. Call its slope $m$ Find the midpoint of the segment by averaging the coordinates of the endpoints The center will be on the line that is perpendicular to the line through the endpoints at the midpoint. Its slope will be $\frac {-1}m$. You have a point and a slope-that gives the equation of the ...



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