(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Differentiate the function

[tex]f(u)=\frac{\ln u}{1+\ln(2u)}[/tex]

2. Relevant equations

- Quotient rule?

- Logarithmic differentiation?

3. The attempt at a solution

I just learned about logarithmic differentiation so I think the idea here was to use logarithmic differentiation.. however, with ln's on the top and bottom, I was unsure of how to go about that and used the quotient rule instead, which got nasty.

[tex]f(u)=\frac{\ln u}{1+\ln(2u)}[/tex]

[tex]f'(u)=\frac{(1+\ln(2u))\frac{d}{du}\ln u-\ln u\frac{d}{du}(1+\ln(2u))}{(1+\ln(2u))^2}[/tex]

[tex]\frac{(1+\ln(2u))\frac{1}{u}-\ln u(\frac{2}{2u})}{(1+2\ln(2u)+(\ln(2u))^2}[/tex]

[tex]\frac{\frac{1+\ln(2u)}{u}-\frac{\ln u}{u}}{(1+2\ln(2u)+(\ln(2u))^2}[/tex]

[tex]\frac{1+\ln(2u)-\ln u}{u(1+2\ln(2u)+(\ln(2u))^2)}[/tex]

The book gives

[tex]f'(u)=\frac{1+\ln2}{u[1+\ln(2u)]^2}[/tex]

Oh my god that was a **** to type up :(

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# Homework Help: Differentiate [ln(u)]/[1+ln(2u)]

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