Expert: Ahmed Salami - 2/27/2006

Question
use logarithmic differentiation to find the derivative of:

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

Answer
Hi Chevonne,
Sorry for the time it took.
For y = [(x^2 + 1)/(x^2 - 1)]^(1/4)
we raise both sides to the power of 4
y^4 = (x^2 + 1)/(x^2 - 1)
Taking the (natural)logarithm of both sides
ln(y^4) = ln[(x^2 + 1)/(x^2 - 1)]
4lny = ln(x^2 + 1) - ln(x^2 - 1)
let u = x^2
du/dx = 2x
4lny = ln(u+1) - ln(u-1)
Differentiating, with respect to u
4(1/y)dy/du = (1/u+1) - (1/u-1)
(4/y)dy/du = [(u-1) - (u+1)]/(u+1)(u-1)
(4/y)dy/du = (u - 1 - u - 1)/(u^2 - 1)
(4/y)dy/du = -2/(u^2 - 1)
(4/y)dy/du = 2/(1 - u^2)
dy/du = y/2(1 - u^2)
But dy/dx = (dy/du).(du/dx)
         = y/2(1 - u^2) . 2x
         = xy/(1 - u^2)
Substituting back u = x^2
dy/dx = xy/(1 - x^4)

I hope i have helped. You can always get back to me.
Regards.