Question

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

Answer 1

`(x(-x^2 +3))/(sqrt(x^2-1)*(x^2+1)^2)`

Explanation:

The derivative can be found by using the quotient rule, which states that if `f(x) = g(x)/(h(x))` then
`f'(x) = (h(x)g'(x) - h'(x)g(x))/(h^2(x))`

In this case `g(x) = (x^2-1)^(1/2)`
and `h(x) = (x^2+1)`

Using the chain rule, `g'(x) = 1/2(x^2 - 1) ^(-1/2)*2x = x/sqrt(x^2-1)`
`h'(x) = 2x`

Then `f'(x) = ((x^2+1)*(x/sqrt(x^2-1)) -2x*sqrt(x^2-1))/(x^2+1)^2`

`=(x(x^2+1) - 2x(x^2-1))/(sqrt(x^2-1)*(x^2+1)^2)`

`=(x^3+x-2x^3+2x)/(sqrt(x^2-1)*(x^2+1)^2) = (x(-x^2 +3))/(sqrt(x^2-1)*(x^2+1)^2)`