Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. $$ 1/x(x+1)(x+2) $$
and I know that $$log((ab)/c) = log(a) + log(b) - log(c)$$ So I tried to use that rule here and did: $$ln(a) - ln(b) - ln(c)$$ and got: $$ln(1) - ln(x(x+1)- ln(x(x+2)$$ which simplifies to: $$0 - ln(x^2+x)- ln(x^2+2x)$$
and then I look at the solution which gives: $$y`=(1/(x(x+1)(x+2))) * (1/x + 1/(x+1) + 1/(x+2))$$
I'm just kind of confused on what I am doing wrong.
You have that
$$\ln f(x)=\ln \frac{1}{x(x+1)(x+2)}=\ln 1-ln x- \ln (x+1)-\ln(x+2)$$ $$=-ln x- \ln (x+1)-\ln(x+2).$$
Taking derivatives:
$$\frac{f'(x)}{f(x)}=-\frac 1x-\frac{1}{x+1}-\frac{1}{x+2}.$$ Thus
$$f'(x)=\frac{-1}{x(x+1)(x+2)}\left(\frac 1x+\frac{1}{x+1}+\frac{1}{x+2}\right).$$
Note a difference in sign with your expected answer.