SOLVED:Differentiate The Function. G(x) = \ln (xe^{-2x}) - Numerade

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Question

Differentiate the function. $ g(x) = \ln (xe^{-2x}) $ Differentiate the function.$ g(x) = \ln (xe^{-2x}) $ Calculus: Early Transcendentals James Stewart 8th Edition Chapter 3, Problem 9 View All Chapters

Step 1

The multiplication inside the logarithm can be separated into two different logarithms using the sum rule. So, we rewrite the function as follows: $$ g(x) = \ln(x) + \ln(e^{-2x}) $$ Show more…

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Transcript

00:01 So in this problem, what we have is y being equal to the natural log of xe to the negative 2x. 00:11 So with this graph, we see that the domains restricted from zero to infinity. 00:16 And the reason why we are doing this problem is because we want to have a better feel of how to take the derivative of special functions, in this case, natural log functions and logarithmic functions in general. 00:29 So what we end up having is that y is equal to we can separate this whenever we have multiplication within the log. 00:39 We can separate this into two different logs using the sum rule. 00:43 So this will be the natural log of x plus the natural log of e to the negative 2x. 00:55 Then since this is a power right here, we can rewrite this as the natural log. 01:04 Make the put the 2x out in front. 01:08 So we'll do a minus 2x right here of the natural log of e... Need help? Use Ace Ace is your personal tutor. It breaks down any question with clear steps so you can learn. Start Using Ace Ace is your personal tutor for learning Step-by-step explanations Instant summaries Summarize YouTube videos Understand textbook images or PDFs Study tools like quizzes and flashcards Listen to your notes as a podcast

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