In such problems try to eliminate square root terms by rearranging and squaring of sides as follows:

Given

√(Y+X) + √(Y-X) = C

Squaring both the sides

(Y+X) + (Y-X) + 2√(Y²-X²)   = C²

or 2√(Y²-X²)   = C² – 2Y

Squaring again

4(Y²-X²)   = (C² – 2Y)²

or 4(Y²-X²)   = (C⁴ + 4Y² -4YC²)

or 4YC² + C⁴ = 4 X²

Differentiating both the sides w.r.t ‘X’ we obtain,

4C².dY/dX= 8X

or dY/dX= 2X/ C²

Differentiating again w.r.t X

d²Y/dX² = 2/ C² (Hence Proved)