What is the derivative of y = ln( x*sqrt (x +5) / (x -1)^3)?  

Expert Answers
justaguide eNotes educator| Certified Educator

We have the function y = ln( x*sqrt (x +5) / (x -1)^3) and we need to find its derivative. Now we first use the properties of logarithms to simplify the expression and make it easier to find the derivative.

We have the relations: log a^b = b* log b , log (a*b) = log a + log b and log(a/b) = log a – log b.

ln( x*sqrt (x +5) / (x -1)^3)

=> ln( x*sqrt (x +5)) – ln (x -1)^3)

=> ln x + ln (sqrt (x +5)) – 3* ln (x -1)

=> ln x + (1/2)* ln (x +5) – 3* ln (x - 1)

The derivative of ln x is 1/x. Using this we get y’ as

1/x + (1/2*(x + 5)) – 3/(x-1)

Therefore the derivative of  y = ln( x*sqrt (x +5) / (x -1)^3) is 1/x + (1/2*(x + 5)) – 3/(x-1).