If $y=\log _e\left(\sin \left(\frac{x^2}{3}-1\right)\right)$ then $\frac{d^2 y}{d x^2}$ is equal to :

$\frac{2}{3} \cot \left(\frac{x^2}{3}-1\right)-\frac{4 x^2}{9} cosec^2\left(\frac{x^2}{3}-1\right)$

$y=\log \left[\sin \left(\frac{x^2}{3}-1\right)\right] \Rightarrow \frac{d y}{d x}=\frac{\cos \left(\frac{x^2}{3}-1\right)}{\sin \left(\frac{x^2}{3}-1\right)} \times \frac{2 x}{3}$

so $\frac{d y}{d x}=\frac{2 x}{3} \cot \left(\frac{x^2}{3}-1\right)$

so  $\frac{d^2y}{dx^2} = \frac{2}{3} \cot \left(\frac{x^2}{3}-1\right)-\frac{2x}{3} × \frac{2x}{3} cosec^2\left(\frac{x^2}{3}-1\right)$

Option: B