Question

What is the derivative of `f(x)=sin^-1(x)` ?

Answer 1

Most people remember this
`f'(x)=1/{sqrt{1-x^2}}`
as one of derivative formulas; however, you can derive it by implicit differentiation.

Let us derive the derivative.
Let `y=sin^{-1}x`.

By rewriting in terms of sine,
`siny=x`

By implicitly differentiating with respect to `x`,
`cosy cdot {dy}/{dx}=1`

By dividing by `cosy`,
`{dy}/{dx}=1/cosy`

By `cosy=sqrt{1-sin^2y}`,
`{dy}/{dx}=1/sqrt{1-sin^2y}`

By `siny=x`,
`{dy}/{dx}=1/sqrt{1-x^2}`