Question

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

Answer 1

`d/dx sin^(-1)x = 1/sqrt(1 - x^2)`

Explanation:

Let `y=sin^(-1)x => siny=x`

Differentiating implicitly we have:

`cosydy/dx=1`
`dy/dx=1/cosy`

And using the identity `sin^2y + cos^2y-=1` we have

`x^2 + cos^2y=1`
`:. cos^2y = 1 - x^2`
`:. cosy = sqrt(1 - x^2)`

Hence, `dy/dx=1/cosy = 1/sqrt(1 - x^2)`