Question

What is the derivative of `arctan sqrt(x^2 -1)`?

Answer 1

Chain rule

Explanation:

`y = tan^(-1)(sqrt(x^2-1))`
`dy/(dx) = 1/(1+(sqrt(x^2-1))^2) *1/(2sqrt(x^2-1)) . 2x`
`= 1/(x.sqrt(x^2-1))`

Another way to solve this is to use different variables and then reduce the equation.
Let `u = sqrt(x^2-1)`, `v=x^2`, then
`y = tan^(-1)u`
`u = sqrt(v-1)`
`dy/(dx) = dy/(du).(du)/(dv).(dv)/(dx)`
`= 1/(1+u^2).(1/(2sqrt(v-1))).2x`
substituting the relationships back we get,`dy/(dx) = 1/(1+(sqrt(x^2-1))^2) *1/(2sqrt(x^2-1)) . 2x`
`= 1/(x.sqrt(x^2-1))`