Victoria V. answered • 04/01/16

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20+ years teaching Calculus

When my students cannot get the right answer, it is usually because the are forgetting the chain rule. And in this problem, you need the quotient rule, and every time you take the derivative in the quotient rule it will involve the chain rule (on only the ln in this problem).

Here is what I get: [(bottom)*(deriv of top) - (top)*(deriv of bottom)]/(bottom)

^{2} [(2x

^{2})*(1/(3x)*3) - (ln(3x)*(4x)]/(2x^{2})^{2}This simplifies to [2x - 4x ln(3x)]/4x

^{4}This further simplifies to [1-2ln(3x)]/2x

^{3}Setting the first derivative = 0 (to find location of mins and/or maxs) get 1 - 2ln(3x) = 0, so 1 = 2ln(3x), so 1/2 = ln(3x), so e

^{1/2}=3x, and x=√e/3There will be a vertical asymptotote at x=0 (x in denom of original problem)

The domain is x>0 (argument of logarithm is always positive as long as working with only real numbers)

Between 0 and √e/3, the first derivative is positive, and after √e/3, the first deriv is negative. This means that there is a maximum at x=√e/3. If you need the point, plug x=√3 back into the original equation and get the maximum at at [√e/3, 9/(4e)] or approximately (0.550, 0.828)

The critical values are the x-values for all mins and maxs AND ALSO all asymptotes and holes.

So the Critical Values are √e/3, and 0.

There is a maximum at √e/3

The value of the maximum is 9/(4e).

Philip P.

ln(3x) is a composite function and requires the chain rule.

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04/01/16

Victoria V.

Thank you, Phillip.

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04/01/16

Victoria V.

To help my students remember to do the chain rule - I have them do it on EVERY derivative, and I will explain this:

Let's say that f(x) = x

^{2}so f'(x) = 2x. Easy enough. I know, you do not NEED the chain rule. But what if you looked at the problem like this:If you treat this as a general power rule, f(x) = (x)

^{2}, then when you take the derivative you get f'(x) = 2 (x)^{1}(1) = 2x. Same thing, but the students see that the chain rule ALWAYS applies, it is just not necessary with a "plain old 'x'".I have had students ask why the chain rule applies to some problems but not others, and so I do the chain rule EVERY time so that they don't forget to do it when it is absolutely necessary.

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04/01/16

Mark M.

^{-2})04/01/16