Logarithmic Function Derivative Simplification Assistance

http://i.imgur.com/OgOEINK.jpg

You can see the question and my answer/method to the question on the image.

However, the section i circled in red is what i am confused about. Would the ln (x-1)^2; which is placed there because we used the quotient rule for the question, cancel out both the ln (x-1) and ln (x+2) in the numerator?

The only reason why i did that was because it is the only way i was lead to the correct answerl

-3 / [(x+2) (x-1)]

Thanks.

ln(a / b) does not equal ln(a) / ln(b), from the property of logs, split the log into two separate logs through subtraction:

y = ln(x + 2) - ln(x - 1)

Now take the derivative of each term:

dy/dx = [1/(x + 2) * 1] - [1/(x - 1) * 1]

dy/dx = 1 / (x + 2) - 1 / (x - 1)

Optionally, you can make the denominators common and combine them into a single fraction:

dy/dx = -3 / (x - 1)(x + 2)