Question

What is the derivative of `y=arccos(x)`?

Answer 1

By Implicit Differentiation,
`y'=-1/sqrt{1-x^2}`.

Let us look at some details.
`y=cos^{-1}x`

by rewriting in term of cosine,
`Rightarrow cos y=x`

by implicitly differentiating with respect to `x`,
`Rightarrow -sin y cdot y'=1`

by dividing by `-sin y`,
`y'=-1/sin y`

by the trionometric identity `sin y =sqrt{1-cos^2y}`,
`y'=-1/sqrt{1-cos^2y}`

by `cos y =x`,
`y'=-1/sqrt{1-x^2}`