00:01
All right, so for this problem, we want to find, we've got an equation here, y equals x plus x to the x, all right, in black here.
00:13
We've got a point, and we're asked to find the tangent, the equation of the tangent line, and the equation of the normal line at that point.
00:28
So what we're going to do is first to find a equation.
00:34
Equation of a line we need a point we need a slope okay we've got our point zero zero nice easy point to work with we need our slope and so if we're looking for a slope one way to find that is to take the derivative okay because the derivative keeps track of the slopes all right so if we take the derivative and then plug in x equals zero that'll tell us the slope at x equals zero in this equation here.
01:10
So we're going to try to find the derivative.
01:12
So let's go ahead and say, all right, y prime is equal to.
01:22
All right, this first term we take the derivative of x.
01:26
Well, the derivative is just one.
01:29
Okay, plus, now we're going to take the derivative of the second term, x times either the x.
01:35
We've got two x's in here.
01:37
So we need to be careful, okay? this is x times times e the x.
01:45
So we're going to use product rule.
01:47
Okay, with, and that is in green up here.
01:50
Okay.
01:51
And so our f is going to be x.
01:57
Our g is going to be e to the x.
02:02
All right.
02:03
And we need f prime and g prime.
02:05
So f prime, the derivative of x is just one.
02:09
G prime, the derivative of e to the x is still e to the x.
02:14
Okay.
02:14
So we've got all the pieces we need now.
02:17
We do f prime, that's just one, times g, that's e to the x.
02:26
All right.
02:28
And just to make sure we're good here, i'm going to put this all in parentheses, okay? and then plus f, that is x, times g prime, that is e to the x.
02:48
All right.
02:52
And it turns out we don't really need those parentheses.
02:54
These.
02:55
So our y prime doesn't need that.
02:59
But there's no need to really clean this up at all anyway, because at this point, we're just going to go ahead and plug in x equals zero.
03:08
Right, x equals zero.
03:11
So if we plug in at x equals zero, this is going to give us our slope of our original equation at that point.
03:26
So at x equals 0, we get 1 plus e to the 0 plus 0 times e to the 0.
03:38
This is 0 times something, that's just 0.
03:41
So we get 1 plus e to the 0...