Question

If \( \sqrt{x + y} + \sqrt{y - x} = 5 \), then \( \frac{d^2y}{dx^2} = \)

• A. \( \frac{2}{25} \)
• B. \( \frac{2}{5} \)
• C. \( -\frac{2}{5} \)
• D. \( -\frac{2}{25} \)

Hint:

For second-order derivatives, first find the first derivative using the chain rule, then differentiate again to obtain the second derivative.

Answer 1

The Correct Option is A

Step 1: Differentiate the equation.
We are given \( \sqrt{x + y} + \sqrt{y - x} = 5 \). Differentiate both sides with respect to \( x \). Use the chain rule: \[ \frac{d}{dx} \left( \sqrt{x + y} \right) + \frac{d}{dx} \left( \sqrt{y - x} \right) = 0 \] Step 2: Solve for \( \frac{dy}{dx} \).
After simplifying, we find: \[ \frac{dy}{dx} = \frac{2}{25} \] Step 3: Conclusion.
The correct answer is (A) \( \frac{2}{25} \).