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Determine the derivative.$$f(x)=x^{2} e^{2 x}-2 x e^{2 x}+4 e^{2 x}$$

$\left(2 x^{2}-2 x+6\right) e^{2 x}$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

Oregon State University

Harvey Mudd College

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all right. There's probably a fast way to do this problem, but there's a lot of product rules in here. Um, or you could factor out from the get go. It's really up to you how you would prefer to do this problem. Eso Maybe I'll just do this the smart way, but you can do the product individually if you wanted to. So what do I mean by the smart way it would be Thio. Identify that you could greatest common factor and eat the two x out of this problem. And so you will be left with X squared minus two X plus forward. If you don't believe me, then distribute that heat of the two X in and you'll get the same answer. That way you can identify that there's just one product instead of doing three different products. Excuse me. So then, when you take the derivative, you say Okay, well, the derivative of bill Aside E to the two x s E to the two x times the derivative up to X, that's the chain will Sometimes, too. That's the derivative of two X. You leave that X squared minus two X plus four alone and then plus, now you leave that he'd to the two x alone and times it by the derivative of the inside, which should be two X minus two. I should be saying that drivel of the right side inside, But then from here, you can actually simplify this further. Now, some teachers will let you leave the answer like this. Otherwise, you could factor out and eat to the two X, And what I'm actually gonna do is distribute that to into the problem. So I'm looking at two X squared minus four X plus eight. That way, when I actually combine like terms, I know I'm not missing anything, because then I could say Okay, well, I have two x squared, Uh, minus two x plus six. Um, and some teachers will let you leave your answer like that. Uh, so let me circle. This is good. Uh, this is good. Or you could even factor attitude out of that. Uh, it is the two x need to be left with X squared minus X plus three. Just dividing everything by two up to you

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