If x ^ 2 + y ^ 2 + y = 4 , then the value of d ^ 2 y d x ^ 2 at the...

Given ${x}^{2} + {y}^{2} + \sin y = 4$

Differentiating both sides with respect to $x$, we get

$2x + \left(2y + \cos y\right)\dfrac{dy}{dx} = 0$

$⇒ \dfrac{dy}{dx} = \dfrac{- 2x}{2y + cos y}$

At \left(- 2, 0\right), \dfrac{dy}{dx} = \dfrac{4}{1} = 4$</p><p>Also, \left(2y + \cos y\right)\dfrac{dy}{dx} + 2x = 0$

Again differentiating with respect to $x$, we get

\left(2y + \cos y\right)\dfrac{{d}^{2}y}{d{x}^{2}} + \left(2 - \sin y\right){\left(\dfrac{dy}{dx}\right)}^{2} + 2 = 0$</p><p>At \left(- 2, 0\right), \dfrac{{d}^{2}y}{d{x}^{2}} + \left(2 - 0\right){\left(4\right)}^{2} + 2 = 0$

$⇒ \dfrac{{d}^{2}y}{d{x}^{2}} = - 34$