Question

find the indicated derivatives
y=ln((2x+5)(5x-2)/(x+1)), y''

Answer 1

This question becomes a lot simpler if we remember some basic properties of logarithms, namely ln(x*y) = ln(x) + ln(y) and ln(y/x) = ln(y) - ln(x).
So we can rewrite our equation as:
\[y = \ln(2x+5) + \ln(5x-2) - \ln(x+1)\]
then
\[y' = 2/(2x+5) + 5/(5x-2) - 1/(x+1)\]
and
\[y'' = [-4/(2x+5)^2] -[25/(5x - 2)^2] + [1/(x+1)^2] \] (after some algebra, and remember to use the chain rule)