00:01
All right, we want to find an equation of the tangent line to this curve here at the point 3 -4.
00:09
So to write an equation of a line, we need a point, which we have, and we need a slope.
00:17
And since they're asking for a tangent, we know that the slope is equal to the derivative at the point, derivative at 3 -4.
00:27
Okay, we can't, well, we could, but it would be a lot of trouble to solve this for why, so we're just going to take the derivative implicitly, which means take the derivative without solving for y.
00:37
The derivative of x squared is 2x, plus the next one is a product.
00:44
I'm going to put the two in front.
00:45
I'm going to use x as the first and y is the second.
00:49
First, derivative of the second, which is y prime, plus the second times the derivative of x, which is one, minus the derivative of y squared, which is two times y times the derivative of y, which is y prime, plus the derivative of x, which is 1, equals the derivative of 20, which is 0.
01:14
All right, so we have 2x plus 2xy prime, plus 2y, minus 2yy, y prime, plus 1 equals 0.
01:26
All right, so what you usually do next is you solve this for y prime, but we're going to plug in the point 3, 4, x is 3, y is 4, so let's just go ahead and plug that in, and then we can find out what y is.
01:40
2 times 3 plus 2 times 3 times y prime, plus 2 times 4 minus 2 times 4 times y prime plus 1 equals 0...