Question

Y=xlog(x-1/x+1) nth derivative solved

Answer 1

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}

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Answer 2

Answer:

Thanks for free points