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Use the properties of logarithms to find the derivative. Hint: It might be easier if you use the properties of logarithms first.$$y=\ln (x \ln x)$$

$$\frac{1+\ln x}{x \ln x}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Missouri State University

McMaster University

Idaho State University

Lectures

01:18

Use the properties of loga…

01:19

01:22

01:06

01:44

Use logarithmic differenti…

01:46

02:15

01:11

02:22

01:14

01:24

eso looking at this problem? Um, I don't know how much of a benefit it would be to use the law of logs, but I'm gonna go ahead and do that, because that's what the directions said to do. Uh, so we have this multiplication inside of this problem so we could rewrite. This is natural log of X plus the natural log of the natural log of X. Well, again, I don't really see the benefit of doing it this way, but I'm following the direction. So if we're then doing the derivative of this, uh, the left, you know, toe left of the plus sign would just be the derivative of that which is one of her ex. Whereas to the right, what we have is one over natural log of X times the derivative of the inside, which is one over X S o. I don't know if anybody would try to simplify that or just say, Hey, this is a good enough answer. Because you have a couple options you could factor out of one over X, and then you'll be left with one plus one over natural log of X. That's an acceptable answer or you could try to get the same denominator, which would be multiplying. Um, the left fraction, by natural log effects, would be natural log of X plus one. But like parentheses, there over X natural log of X, which I think is actually how the answer key is written. Um, And if you did that way, you probably would have done the chain rule from the beginning, which would have been the derivative of that would be one over X natural log of X times. The derivative of the inside would be the chain rule, uh, or site product rule inside of this, where the derivative of excess one, you leave natural log of X alone. Plus, now you leave X alone. The drift of of natural objects is one over X, and that does simplify to that answer down here. So again there's there's lots of different ways of writing this. I'll circle all of them and you choose. There you go

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