Question

y = x log ((x - 1)/(x+1))​

Answer 1

Answer:

Given : y = x ln{(x - 1)/(x +1)}

To Find : nth derivative

Solution:

y = x ln{(x - 1)/(x +1)}

log (a/b) = loga - logb

=> y = x {ln (x - 1) - ln(x + 1) }

=> y = x ln(x - 1) - x ln(x + 1)

=> y' = x /(x - 1) + ln(x - 1) - x/(x + 1) - ln(x + 1)

=> y' = x/(x - 1) - x/(x + 1) + ln(x - 1) - ln(x + 1)

=> y' = (x - 1 + 1)/(x - 1) - (x + 1 - 1)/(x + 1) + ln(x - 1) - ln(x + 1)

=> y' = 1 + 1/(x - 1) - 1 + 1/(x + 1) + ln(x - 1) - ln(x + 1)

=> y' = 1/(x - 1) + 1/(x + 1) + ln(x - 1) - ln(x + 1)

Now nth Derivative can be found

yₙ = (-1)ⁿ⁻¹(n-1)!/(x - 1)ⁿ +(-1)ⁿ⁻¹(n-1)!/(x+ 1)ⁿ + (-1)ⁿ⁻²(n-2)!/(x - 1)ⁿ⁻¹ -(-1)ⁿ⁻²(n-2)!/(x+ 1)ⁿ⁻¹

yₙ = (-1)ⁿ⁻²(n-2)! { -1(n-1)/(x - 1)ⁿ -1(n-1)/(x + 1)ⁿ + 1/(x - 1)ⁿ⁻¹ - 1/(x+ 1)ⁿ⁻¹ )

yₙ = (-1)ⁿ⁻²(n-2)! { -1(n-1)/(x - 1)ⁿ -1(n-1)/(x + 1)ⁿ + (x-1)/(x - 1)ⁿ - (x+1)/(x+ 1)ⁿ )

=> yₙ = (-1)ⁿ⁻²(n-2)! {(x-n)/(x - 1)ⁿ -(x + n)/(x + 1)ⁿ}