Question

y= 4th root (x^2+1)/(x^2-1)

Answer 1

y = [(x^2 + 1)(x^2-1) ]^(1/4)
= (x^4 - 1)^(1/4)
ln y = (1/4) ln(x^4 -1)
(dy/dx) / y = (1/4)(4x^3)/(x^4 - 1)
dy/dx = (1/4)(y)(4x^3)/(x^4 - 1)
or
you can replace y with (x^4 - 1)^(1/4)
and simplify that a bit since the denominator is x^4 - 1

Answer 2

Do you want the derivative of that? Does the "fourth root" apply to the numerator only, or the complete fraction
(x^2+1)/(x^2-1) ?

You need to write the function in a clear nonambiguous manner, using parentheses where necessary. Use ^1/4 for fourth roots.

Answer 3

Sorry ana, my answer is incorrect,
I read that as a multiplication , should have been a division

ln y = (1/4) (ln (x^2 + 1) - ln(x^2 -1)

(dy/dx) / y = (1/4) ( 2x/(x^2+1) - 2x/(x^2 + 1) )
dy/dx = (1/4)(y) (4x)/(x^4 - 1)
= xy/(x^4 - 1)

replace y with the original if you have to.