00:01
So here we are given y is equal to x y squared plus 2 x squared.
00:05
And we want to find the equation of all the tangent lines when x is equal to negative 1.
00:11
So let's go ahead and plug in x equals negative 1 to figure out what our y values are equal to.
00:17
So plug in negative 1, we get y equals negative y squared plus 2.
00:22
So when we get all our y's and numbers on the same side, we get y squared plus y minus 2 equals 0.
00:29
And this factors to become y plus 2 times y minus 1 equals 0.
00:35
And when we solve for y, we get y equals negative 2 and positive 1.
00:40
So those are the two y values we get when x is negative 1.
00:45
So that's how and also negative 1.
00:52
Next, we want to find the derivative.
00:56
So because we have y's and x's, we're going to need to use the implicit differentiation.
01:01
So for implicit differentiation, we're going to get that y prime is equal to, and then here we have a product rule.
01:10
So we're going to have x times 2y times y prime and then plus y squared.
01:18
And then we have a plus 4x.
01:20
So moving this term to the left side, we're going to have y prime minus 2xy y prime is equal to y squared plus 4x.
01:30
We can factor out a y prime to get 1 minus 2xy is equal to y squared plus 4x.
01:38
And then we can divide both sides by that 1 minus 2xy to get y squared plus 4x all over 1 minus 2xy.
01:47
So now we have our derivative of y and we know our x and y values...