Question

If $x^2+y^2+\sin y=4$, then the value of $\frac{d^{2}y}{d{x}^2}$ at the point $(-2,0)$ is

• A. $-34$
• B. $-32$
• C. $-2$
• D. $4$

Answer 1

The correct option is A $-34$

Given,

$x^2+y^2+\sin y=4$

On differentiating the equation the above equation w.r.t $x$, we get

$2x+2y\frac{dy}{dx}+cosy\frac{dy}{dx}=0...\left(i\right)$

Again differentiating equation $\left(i\right)$ w.r.t to $x$, we get

$2+2(\frac{dy}{dx}{)}^2+2y\frac{d^{2}y}{d{x}^2}-\sin y(\frac{dy}{dx}{)}^2+\cos y\frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow 2+(2-\sin y)(\frac{dy}{dx}{)}^2+(2y+\cos y)\frac{d^{2}y}{d{x}^2}=0$

$\Rightarrow (2y+\cos y)\frac{d^{2}y}{d{x}^2}=-2-(2-\sin y)(\frac{dy}{dx}{)}^2$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{-2-(2-\sin y)(\frac{dy}{dx}{)}^2}{2y+\cos y}$

Therefore, at (-2,0),

$\frac{d^{2}y}{d{x}^2}=\frac{-2-(2-0)\times 4^2}{2\times 0+1}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=\frac{-2-2\times 16}{1}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=-34$

Hence, correct option is $\left(A\right)-34$