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Write $x^{x}$ as $e^{x \ln x}$ to find its derivative. (See Example 16 .)

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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$$\text { } , \text { …

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Find the derivative of the…

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Okay, So the directions want you to find the derivative of X to the X, And it is true that X of the X is equal thio e to the X natural log of X. If you're confused by that directions because, um, you know, this ex city could be written as a power, and then the E and the natural log would cancel out and you'd be left with X to the XO. They gave you that information. That's not really important. What's important now is that we're going to take the derivative of that. So, like, we're gonna find the derivative of this by using this rule. Yeah, eso Let's let's go ahead and attack the problem. Well, the derivative of e tow. Any power is itself. But what's important is that you take the drift of of this piece. Well, that's the product rule. So times by, uh, the derivative of X is one. You leave natural log of x alone. Plus, now you leave X alone in the derivative of natural log of X is one over x. So as I clean this up a little bit, I'll go back to read is like I said, a to the X natural log of X is itself times the quantity of natural law. Globex because one times, anything is is that thing. And then x times one of the wrecks cancels each other out. So plus one, it might be important that you put, you know, in X right there in parentheses. But remember the very beginning, we said x to the X is equal thio e to the X natural log of X. So this piece right here is actually equal to X to the X. Like I said at the beginning, and you can actually leave this alone Natural log of X plus one or some people wolf. Sorry. Distribute in this X to the X. Um, but you don't have to. Maybe I will. Just just for the sake of writing something X else X t x times natural Aga, Becks plus X to the X. Anyway, either one of these answers is correct. I'm circling in both and green. It's up to you What you want to write down

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