y= 4th root (x^2+1)/(x^2-1)
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
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.
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.
To simplify the expression \(y = \sqrt[4]{\frac{{x^2+1}}{{x^2-1}}}\), we can break it down into smaller steps.
Step 1: Simplify the expression inside the fourth root.
The expression inside the fourth root is \(\frac{{x^2+1}}{{x^2-1}}\). To simplify this, we can factor the denominator using the difference of squares formula.
\(x^2-1\) can be written as \((x+1)(x-1)\).
So, the expression becomes \(\frac{{x^2+1}}{{(x+1)(x-1)}}\).
Step 2: Simplify further.
Now that we have the expression \(\frac{{x^2+1}}{{(x+1)(x-1)}}\), we can simplify it by canceling out common factors.
The numerator, \(x^2+1\), cannot be simplified any further.
In the denominator, we have \((x+1)(x-1)\). There are no common factors to cancel out.
Therefore, the simplified expression is \(\frac{{x^2+1}}{{(x+1)(x-1)}}\).
Step 3: Take the fourth root.
Finally, the expression \(y = \sqrt[4]{\frac{{x^2+1}}{{(x+1)(x-1)}}}\) means taking the fourth root of the simplified expression.
To take the fourth root, we raise the expression to the power \(1/4\).
Therefore, the final expression is \(y = \left(\frac{{x^2+1}}{{(x+1)(x-1)}}\right)^{1/4}\).
Now you have the simplified expression for \(y\) in terms of \(x\).