If y = √(2024x+2025) , then which of the following is correct?

The correct answer is Option (4) → \(y\displaystyle\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^2=0\)

Squaring first removes the root, and then implicit differentiation is short:

    \(y^2 = 2024x + 2025\)

Differentiating with respect to x:

    \( \displaystyle 2y\frac{dy}{dx} = 2024 \)

Differentiating again, by the product rule:

    \( \displaystyle 2\left(\frac{dy}{dx}\right)^2 + 2y\frac{d^2y}{dx^2} = 0 \)

    \( \displaystyle y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 = 0 \)