Answer:
y" = -24 / y³
Explanation:
6x² + y² = 4
Take the derivative of both sides with respect to x.
12x + 2y y' = 0
Again, take the derivative of both sides with respect to x.
12 + 2y y" + y' (2y') = 0
12 + 2y y" + 2(y')² = 0
Solve for y' in the first equation.
2y y' = -12x
y' = -6x/y
Substitute and solve for y":
12 + 2y y" + 2(-6x/y)² = 0
12 + 2y y" + 2(36x²/y²) = 0
12 + 2y y" + 72x²/y² = 0
6y² + y³ y" + 36x² = 0
y³ y" = -36x² − 6y²
y" = (-36x² − 6y²) / y³
Solve for y² in the original equation and substitute:
y² = 4 − 6x²
y" = (-36x² − 6(4 − 6x²)) / y³
y" = (-36x² − 24 + 36x²) / y³
y" = -24 / y³