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Differentiate the function.
$ g(x) = \ln (xe^{-2x}) $
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00:18
Frank Lin
00:45
Doruk Isik
Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 6
Derivatives of Logarithmic Functions
Derivatives
Differentiation
Blake C.
October 3, 2020
Missouri State University
Oregon State University
Idaho State University
Boston College
Lectures
04:40
In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
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Differentiate the function…
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Find the derivative of the…
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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.
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