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. $4$
• D. $-2$

Answer 1

The correct option is A $-34$

$x^2+y^2+\sin y=4\Rightarrow 2x+2y\cdot y^{\prime }+y^{\prime }\cdot \cos y=0\Rightarrow y^{\prime }=-\frac{2x}{2y+\cos y}$
Putting $(-2,0)$
$y^{\prime }{|}_{(-2,0)}=4$
Now again differentiating we get,
$y^{\prime \prime }=-\frac{(2y+\cos y)\cdot 2-2x\cdot (2y^{\prime }-y^{\prime }\cdot \sin y)}{(2y+\cos y{)}^2}$
Putting $(-2,0)$
$y^{\prime \prime }{|}_{(-2,0)}=-\frac{2+4\cdot (8-0)}{1}=-34$