Question

How do you differentiate `f(x) = arctan(x^2-1)^(1/2) + arcsec(x)`?

Answer 1

`2/(xsqrt(x^2-1))`

Explanation:

To make f real, `x in [-1, 1]`.

Let `x = sec theta in [-1, 1]`

Now,

`f = arc tan(tan theta)+arc sec ( sec theta)`

`=theta +theta = 2 theta = 2 sec^(-1)x`

So, `f'=2(sec^(-1)x)'=2/(xsqrt(x^2-1))`