−2
4
−2
−4
Given, x² + y² + siny = 4
On differentiating the equation the above equation w.r.t x, we get
2x + 2y(dy/dx) + cosy(dy/dx) = 0.. (i)
Again differentiating equation (i) w.r.t to x, we get
2 + 2(dy/dx)² + 2y(d²y/dx²) - siny(dy/dx)² + cosy(d²y/dx²) = 0
⇒ 2 + (2 - siny)(dy/dx)² + (2y + cosy)(d²y/dx²) = 0
⇒ (2y + cosy)(d²y/dx²) = -2 - (2 - siny)(dy/dx)²
⇒ d²y/dx² = [-2 - (2 - siny)(dy/dx)²] / (2y + cosy)
Therefore, at (-2, 0),
d²y/dx² = [-2 - (2 - 0) × 4] / (2 × 0 + 1)
⇒ d²y/dx² = -10 / 1
⇒ d²y/dx² = -4
From (i), at (-2,0)
2(-2) + 2(0)(dy/dx) + cos(0)(dy/dx) = 0
-4 + dy/dx = 0
dy/dx = 4
Hence, correct option is (D) -4