역삼각함수의 미분

The function $\dfrac{1}{\sqrt{x^2-1}}$ exists only for those values of $x$ with $x^2-1>0$: that is, the domain of $\dfrac{1}{\sqrt{x^2-1}}$ is $|x|>1$. However, the domain of arcsine is $|x| \leq 1$. So, there is not one single value of $x$ where $\arcsin x$ and $\dfrac{1}{\sqrt{x^2-1}}$ are both defined.

If the derivative of $\arcsin(x)$ were given by $\dfrac{1}{\sqrt{x^2-1}}$, then the derivative of $\arcsin(x)$ would not exist anywhere, so we would probably just write “derivative does not exist," instead of making up a function with a mismatched domain. Also, the function $f(x)=\arcsin(x)$ is a smooth curve–its derivative exists at every point strictly inside its domain. (Remember not all curves are like this: for instance, $g(x)=|x|$ does not have a derivative at $x=0$, but $x=0$ is strictly inside its domain.) So, it's a pretty good bet that the derivative of arcsine is not $\dfrac{1}{\sqrt{x^2-1}}$.