Question

Referring to question 5A-3f, why is d/dx(arcsin(a/x)) = -a/(x(((x^2)-(a^2))^(1/2))?

Answer 1

Here is a derivation by Khan, who is a good explainer:
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/derivatives-inverse-functions/v/derivative-inverse-sine

to follow this, you must understand implicit differentiation:
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/implicit_differentiation/v/implicit-differentiation-1

and of course how to take the derivative of sin x
https://www.khanacademy.org/math/differential-calculus/taking-derivatives/chain_rule/v/derivatives-of-sin-x-cos-x-tan-x-e-x-and-ln-x