Wikipedia:Reference desk/Archives/Mathematics/2012 March 9

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March 9

Logarithm problems

I'm having some trouble with a calculus assignment over differentiation of logarithms (a scan of the assignment on the right). The instructions for 24 were to differentiate the function, for 28 they were to find the first and second derivative, and for 44 and 52 they were to differentiate using logarithmic differentiation. I normally use Wolfram Alpha to double check my answers, but my answers are pretty different from what it's telling me and the steps don't even look like what I did. Can anyone tell me where I'm going wrong? (And I'm afraid I don't know for sure what the right answers are because only odd numbered questions have the answers in the back of the book). Thanks, Ks0stm ^{(T•C•G•E)} 00:48, 9 March 2012 (UTC)

Well, I didn't check all four exercises, but it looks like in at least some cases (yes, it's an unrepresentative sample) you can simplify those expressions... --Martynas Patasius (talk) 02:42, 9 March 2012 (UTC)

With '24' you can divide top and bottom to get rid of $\exp(-x)$ then divide top and bottom by $\cos(mx)$ to get $-{\frac {1+m\tan(mx)}{\ln(2)}}$, which is what Wolfram Alpha gives me. 213.249.185.46 (talk) 11:10, 10 March 2012 (UTC) — Preceding unsigned comment added by 213.249.185.46 (talk) 11:08, 10 March 2012 (UTC)

With '28' you've gone off course when using the 'product rule' to differentiate. Write it as:

$y=\ln(x)\cdot {\frac {1}{x^{2}}}$,

then differentiate using the product rule:

${\frac {dy}{dx}}={\frac {1}{x}}\cdot {\frac {1}{x^{2}}}+\ln(x)\cdot {\frac {-2}{x^{3}}}$ .

Add the two bits together (as they have a common denominator of x^3 and I get the same answer as Wolfram Alpha. 213.249.185.46 (talk) 11:16, 10 March 2012 (UTC)

With '44' you expand y perfectly so that it looks like:

$y=-x+\ln(cos^{2}(x))-\ln(x^{2}+x+1)$

but then you must have got confused when differentiating as you get a \exp(-x) from somwhere! Just differentiate term by term so that you get:

$y^{\prime }=-1-2\tan(x)-{\frac {2x+1}{x^{2}+x+1}}$

which is part of your answer - so were on the right track! Wolfram Alpha simplifies this by writing -1 as $-{\frac {x^{2}+x+1}{x^{2}+x+1}}$ and adding it to $-{\frac {2x+1}{x^{2}+x+1}}$ to give $-{\frac {x^{2}+3x+2}{x^{2}+x+1}}$ which can be factorised 213.249.185.46 (talk) 11:27, 10 March 2012 (UTC)

For the last problem you make a mistake when differentiating, you correctly have:

$\ln(y)=\ln(x)\cdot \ln(\sin(x))$, differentiate to get:

${\frac {y^{\prime }}{y}}={\frac {1}{x}}\cdot \ln(\sin(x))+\ln(x)\cdot \cot(x)$, then subs in y(x). Hope this helps, Chris 213.249.185.46 (talk) 11:39, 10 March 2012 (UTC)