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Determine the equation of the tangent line at the given $x$ -coordinate.$$f(x)=x^{3} e^{3 x}, x=1$$

$y=e^{3}(6 x-5)$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

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So when you see the directions asked for the tangent line, just make sure that you find the point and the slope. Yeah, and, uh, as far as the directions go, they only give you the X coordinate of the point. So you have to figure out the y coordinate, and then the slope is found by finding the derivative and plugging in one. And for X eso, you have a little bit of work to do. So first of all, identify what the point is. Uh, well, I guess I should mention the equation Is X cubed times E to the three X Oh, so at least plugging in one. And for all these exes, I think it's pretty straightforward that you recognize that you'll get one times e to the three times one, which would just simplify it to E cube if you wanna leave your answer like that. But then when you have to do is find the derivative, which is your product rule. So find your product, take the derivative of the left side, leave the right side alone, plus leave the left side alone X cubed, and then the derivative of the right side is e to the three X but it's a change A roll. So you have to multiply by the derivative of three X, which is three now based off of looking at the answer key, it looks like they've factored out three x squared eat of the three X on both of those on their left with one plus X. So it makes sense when you plug in one. And for all of those pieces, um, to do f prime of one. Uh, Okay, well, you're plugging one. And for all these exes, eso one plus one is two times this three is six. One squared is still one e two of the three times one. Something like that. Yeah. So how do we get to our final answer is, uh I like to write in point slope form, um, where I take the slope and then x minus the x coordinate, plus the y coordinate. And this is a good enough answer. Otherwise, what you could do is distribute this six e Cube. Ah, into both those terms. So you're looking at six e. That doesn't look like me. Okay. Sorry about that. And then when you distribute six into this. You get negative. Six plus one would be negative. Five Heat a third. Really, There's no benefit to doing to simplifying that, but there you go.

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