Logarithmic differentiation challenge

[lookagain]

\(\displaystyle f(x) \ = \ ln[(x - 7)^5(x + 2)^4(x + 92)^2] \)

Or write "log" for the natural logarithm, if you prefer, instead of "ln."

(i) Work out a version of the solution of the first derivative of f(x).

(ii) Using the answer to the first part, calculate the slope of the
tangent line to the graph of f(x) at x = 8 as a decimal number.

Please use spoiler/hide tags for your work.

 

[Steven G]

As easily seen you can read off the derivative at x=8.
y'=5/(x-7) + 4/(x+2) + 2/(x+92). y'(8) = 5.42

 

[HallsofIvy]

Do you not know that \(\displaystyle ln[(x- 7)^5(x+ 2)^4(x+ 92)^2]= 5ln(x- 7)+ 4ln(x+ 2)+ 2ln(x+ 92)\)?

That together with the fact that the derivative of \(\displaystyle ln(x+ a)\) is \(\displaystyle \frac{1}{x+ a}\) makes this almost trivial!

 

[lookagain]

Do you not know that \(\displaystyle ln[(x- 7)^5(x+ 2)^4(x+ 92)^2]= 5ln(x- 7)+ 4ln(x+ 2)+ 2ln(x+ 92)\)?

Why would you post this question when taken together it is posted as
1) a "challenge," 2) it's in "Math Odds & Ends," 3) it's posted by an Elite Member,
and 4) I asked for spoiler/hide tags in context of the problem?

I also posted it, because there was a similar problem to this of my part a), but the
majority of solvers were not taking advantage of what you and Jomo showed.
I wanted to encourage this method by tacking on part b).