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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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).

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