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Determine the equation of the tangent line at the given $x$ -coordinate.$$f(x)=e^{2(x-3)}, x=3$$

$y=2 x-5$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 4

The Derivative of the Exponential Function

McMaster University

Harvey Mudd College

Baylor University

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uh, we have another tangent line problem, so that means we need a point and they give us the exporting of the point. So he said to figure out the Y coordinate and we need the slope. And since they gave us the X coordinate, what was actually was three e I used to doing a bunch of zeros. The slope is the derivative when X equals three. So as we look at the problem and how they give us that ffx is defined, uh, as e to the two quantity of X minus three, Uh, it actually makes perfect sense that when you plug in three right here to find of three, this is the white coordinates that we end up getting e to the two times zero power because three, minus 30 which is equal to eat of the zero, which is equal to one. Um, but now what we have to do is find the derivative because I will tell us the equation of the change a line. So if you notice what I would probably do is actually distribute that in, uh, would make my life a little bit easier. You know, I have to do it my way. Um, but the reason I think of that is the derivative of that. The explosion is too. Um and then I'm gonna leave the expo alone for the derivative of the So what's nice about that switch back to green is f prime of three is basically two times ffx, which we found out earlier that f of three was equal toe one. So the slope at that point was to So now I'm ready for the equation in point slope form. I like to do the problem this way. Where you write down the slope X minus the X coordinate and then plus the y coordinate. And I let my students leave their answers like this. Otherwise you could distribute that to in there. So you're looking at why equals two X That'll be negative six plus one to make that negative five. This is an acceptable answer as well.

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