Question

Differentiating a Logarithmic Function In Exercise, find the derivative of the function.

y =In x(x - 1)/x - 2

Answer 1

`(d)/(dx) ln((x(x-1))/(x-2)) = (x-2)/(x(x-1)) (x^2 -4x +2)/((x-2)^2) = (x^2 -4x +2)/(x(x-1)(x-2))`

Explanation:

For this case we want to find the derivate of this function:

`y = ln((x(x-1))/(x-2))`

And in order to find the derivate we need to apply the chain rule given by:

`(df(u))/(dx) =(df)/(du) (du)/(dx)`

And on this case
`f = ln(u), u = (x(x-1))/(x-2)`

And we can find the partial derivates like this:

`(d)/(du) (ln(u))= (1)/(u)`

`(d)/(dx)((x(x-1))/(x-2))= ((2x-1)(x-2) -x(x-1))/((x-2)^2)= (x^2 -4x +2)/((x-2)^2)`

And if we replace we got:

`(d)/(dx) ln((x(x-1))/(x-2)) = (1)/(u) (x^2 -4x +2)/((x-2)^2)`

And if we replace
`u = (x(x-1))/(x-2)` we got:

`(d)/(dx) ln((x(x-1))/(x-2)) = (x-2)/(x(x-1)) (x^2 -4x +2)/((x-2)^2) = (x^2 -4x +2)/(x(x-1)(x-2))`

And that would be our final answer on this case