Question

y^3=x^2+y^2 (2nd order derivative )

Answer 1

$\bold{Answer :}$

Given that,

y³ = x² + y²

Now, differentiating both sides with respect to x, we get

d/dx (y³) = d/dx (x² + y²)

⇥ 3y² dy/dx = 2x + 2y dy/dx

⇥ (3y² - 2y) dy/dx = 2x

Again, differentiating both sides with respect to x, we get

d/dx {(3y² - 2y) dy/dx} = d/dx (2x)

⇥ (dy/dx) {d/dx (3y² - 2y)} + (3y² - 2y) d/dx (dy/dx) = 2

⇥ (dy/dx) (6y dy/dx - 2 dy/dx) + (3y² - 2y) d²y/dx² = 2

⇥ (6y - 2) (dy/dx)² + (3y² - 2y) d²y/dx² = 2

⇥ (6y - 2) {2x/(3y² - 2y)}² + (3y² - 2y) d²y/dx² = 2

⇥ d²y/dx² = - [(6y - 2) {2x/(3y² - 2y)}² - 2]/(3y² - 2y)

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