Question

What's the derivative of `sqrt[arctan(x)]`?

Answer 1

`1/(2(x^2+1)sqrtarctan(x)`

Explanation:

This can be written as `(arctan(x))^(1/2)`.

This is in the form `u^(1/2)`. The chain rule with the power rule tells us that:

`d/dxu^(1/2)=1/2u^(-1/2)*u'=1/(2sqrtu)*u'`

So for `arctan(x)`, whose derivative is `1/(x^2+1)`, this becomes

`d/dxsqrtarctan(x)=1/(2sqrtarctan(x))*1/(x^2+1)`