What differentiation rule do you use to differentiate arcsin (1/x)?

Question

I know that arcsin x is 1/sqrt(1-x^2).

Answer 1

Chain rule.

f(x) = arcsin(1/x) = arcsin(y(x)), where y(x) = 1/x

f'(x) = d/dy(arcsin(y)) * d/dx(y(x)) = 1/sqrt(1-y^2)*-1/x^2 = -1/sqrt(1-(1/x)^2)*1/x^2

= 1/x*1/(sqrt(x^2-1))

Answer 2

d/dx(arcsin(u)) = [1/(sqrt(1-u^2)] du/dx