00:03
We're given the equation x cubed plus y cubed equals 16, and we're trying to find the value of the second derivative of y with respect to x at the point to 2 .2.
00:14
So we need to start with the first derivative.
00:17
Taking the derivative here with respect to x, that will give us 3x squared plus 3 y squared times the derivative of y with respect to x, y prime, equals derivative of a constant is 0.
00:35
Now to find our second derivative, we need to solve this for y prime first.
00:41
So we need to isolate the y prime term and then do some dividing.
00:47
So that means that 3 y squared times y prime is going to equal negative 3 x squared.
00:57
And then when we divide both sides by the 3 y squared, we get y prime equals negative x squared over y squared.
01:06
Now, we need to find the second derivative by taking the derivative of this using the quotient rule.
01:14
And i'm going to keep the negative with my numerator here as i'm doing this.
01:20
So that means we're going to have the bottom function, y squared, times the derivative, sorry, that's y double prime.
01:28
So the second derivative is going to equal y squared times the derivative of the negative x squared.
01:35
So that would be negative 2x minus the top, negative x squared, times the derivative of the bottom, which is going to be 2y times y prime...