You did just fine and your second derivative is correct, if you meant (6y^2 - 4x^2)/(9y^3)
except they took it a bit further.
notice your numerator is
-4x^2 + 6y^2
from the original
2x^2 - 3y^2 = 4 , then
4x^2 - 6y^2 = 8 , and
-4x^2 + 6y^2 = -8
to get their -8/9y^3
If you were marked "wrong", then that is bad,
when differentiating implicitly, there are often multiple variations of the same answer.
Here is a way to check if two possible answers are equivalent:
Pick any point which satisfies the original equation, sub that point into the two variations of the derivatives, you should get the same answer if they are equivalent.
e.g.
the point (√8,2) is on the original curve
y'' -- your answer -- = (24 - 32)/72 = -8/72 = -1/9
y'' -- their answer = -8/72 = -1/9
ok, then !!
When you wrote
2x^2 - 3y^2 = 4 , then
4x^2 - 6y^2 = 8 , and
-4x^2 + 6y^2 = -8
Did you just run a ratio for the first two equations? And, then for te third since it was negative, then you just took the negative number for it 8 to negative 8?
And, I was not marked wrong for it- it was a hw problem and the answer in the back had -8 but had gotten the -4x^2 + 6y^2 expression.
Also, how were you able to get the point (√8,2)as a point on the curve? Did you just solve for y? Thanks so much for your help.
Yes, you are correct. To obtain the third equation, they multiplied the second equation by -1. This is a common step when manipulating equations to solve for different variables.
Regarding the point (√8, 2), it is obtained by solving for y in the original equation 2x^2 - 3y^2 = 4. By rearranging the equation and isolating y, you get:
-3y^2 = 4 - 2x^2
y^2 = (4 - 2x^2)/3
y = ±√[(4 - 2x^2)/3]
To find a specific point on the curve, you can substitute a specific value for x into this equation to calculate the corresponding y-coordinate. In this case, (√8, 2) was chosen as an example to check the two variations of the derivative.