Question

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

Answer 1

The derivative of `y=arctan x` is `y'=1/{1+x^2}`.

We can derive this by using implicit differentiation.

Since inverse tangent is hard to deal with, we rewrite it as
`tan(y) =x`

By implicitly differentiating with respect to `x`,
`sec^2(y)cdot y'=1`

By solving for `y'` and using `sec^2(y)=1+tan^2(y)`,
`y'=1/{sec^2(y)}=1/{1+tan^2(y)}`

Hence, `y'=1/{1+x^2}`.