设 \(y = f(x)\) 的反函数 \(g(x)\) 则有 \(x = g(y)\)
\(g'(y) = \frac{d g(y)}{dy} = \frac{dx}{dy} = \frac{1}{f'(x)}\)
设 \(y = arcsin(x)\)，则有 \(x = sin(y)\)
\(arcsin'(x) = \frac{d arcsin(x)}{dx} = \frac{dy}{dx}\)
\(\frac{dx}{dy} = \frac{dsin(y)}{dy} = cos(y)\)
\(arcsin'(x) = \frac{1}{cos(y)}\)
\(\because sin(y) = x\)
\(\therefore cos(y) = \sqrt{1 - x^2}\)
\(\therefore arcsin'(x) = \frac{1}{\sqrt{1 - x ^ 2}}\)