Problem 9 Easy Difficulty

Differentiate the function.

$ g(x) = \ln (xe^{-2x}) $

Answer

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00:18

Frank L.

00:45

Doruk I.

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Video Transcript

so in this problem we have is why being equal to the natural log of X E to the negative two x Just so With this graph, we see that the domains restricted from zero to infinity and the reason why we're doing this problem is because we wanna have a better feel of how to take the derivative of special functions. In this case, um, natural log functions and live arrhythmic functions in general. So what we end up having is that why is equal to weaken? Separate this whenever we have multiplication within the log weaken separate this into two different logs using the some rule. So it will be the NAFTA log of X plus the natural log of e to the negative. Just Dex then, since this is a power right here, we can rewrite this as theme Natural log. We can make it put the two x two x out in front so we'll do. Ah, minus two acts right here of the natural log e. We know that the natural log of e is just one so end up getting natural log of acts minus two X. Then, with this in mind, we can now take the derivative. So why Prime will end up becoming one over acts minus two. That will be our final answer. Another way that we contest This is by looking at the fact that one X one overact my next to is this graph right here and then if this right here was a function of X and F crime of neck or F prime of X when then being this portion of the graph right here Ah, this portion of the graph we see is the derivative of the graph eso We know that we did this properly. This would be our final answer for the derivative. It requires us to simplify the log first and then take the derivative. So it's important that we do that first.