00:01
Hello, we are given this implicit equation for y and we have to find y triple prime at x equal to 1.
00:12
So first let us find out the value of y at x equal to 1.
00:16
So let us plug in x equal to 1 into this equation and see what we get.
00:21
We get 1 plus 1 times y is y plus y cube is equal to 1 and we can cancel the 1.
00:31
We will have y plus y cube equal to 0.
00:33
If you take y common, we have y times 1 plus y square equal to 0.
00:38
Now 1 plus y square factor is always greater than equal to 1 because y square is non -negative.
00:45
So this is never 0.
00:47
So we have that the only solution of this equation is y equal to 0.
00:51
So we see that when x equal to 1, y equal to 0.
00:57
Now let us differentiate both sides of this equation.
01:00
We will get 2x plus 1 times y plus x times y prime plus 3 y square times y prime equal to 0.
01:11
Now into this, let us plug in x equal to 1 and y equal to 0.
01:18
Then this will give us the value of y prime at this point.
01:24
Now when we do that, we get 2.
01:30
This is 0.
01:33
This thing is y prime and this thing is 0.
01:40
Sorry.
01:41
Yeah, we are doing it here.
01:43
So 2 times 1 into this, if you plug in x equal to 1, y equal to 0, we get 2 times 1 plus 0 plus x times y prime, which is 1 times y prime, which is y prime plus 0 equal to 0.
02:01
So 2 plus y prime equal to 0, y prime equal to minus 2.
02:05
So we find that the value of y prime at this point x equal to 1 is minus 2...