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Use logarithmic differentiation to find the derivative.$$y=x^{e^{x}}$$

$$e^{x}(1+x \ln x) x^{e^{x}-1}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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02:15

Use logarithmic differenti…

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eso There's a couple different ways of doing this problem. I would do this problem by taking the natural log on both sides. So, uh, well, that's just write that down first, why equals X E to the X? So then I can take the natural audible sides. The reason why that's beneficial is now I can move this e to the X in front eso. What we have is natural. Log of y is equal to e to the X times natural log of X, and what I can do is take the derivative on the left side, which is implicit differentiation. One over Why D Y d X And on the right side we have the product rule. Well, the drift of of either the X is e to the X. You'll leave natural Lagerback's alone and then plus, you'll leave e to the X alone and the derivative of natural log of X is one over X. Um, now, if you really wanted to, But you don't have to. You could factor out and e to the X on either side. I'm not going to. All right, So what I would do next is you have to solve for D Y d X, where you could do that by multiplying. Why over you might be sitting there and saying What is why? Why is equal to this X e to the X? Um so what we have all right, this in green x e to the X. So it's yeah. So all of this in parentheses is the same. If, um, but then you could rewrite. Why is X e to the X power? And that's what D y d. X is equal to. And if you wanted to, you could factor out. She looks like is what they did and distribute in. Um, you can combine like terms. Looks like there's a lot of things you could do, but this is good enough. We are done. We're good. Thank you.

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