(a)OR(b)

Differentiate $\displaystyle \sec ^{-1}\left(\frac{1}{\sqrt{1-\mathrm{x}^{2}}}\right)$ w.r.t. $\displaystyle \sin ^{-1}\left(2 \mathrm{x} \sqrt{1-\mathrm{x}^{2}}\right)$.

Let $\displaystyle \mathrm{x}=\sin \theta$. Then\begin{aligned} & \mathrm{y}=\tan \mathrm{x}+\sec \mathrm{x}=\frac{\sin \mathrm{x}+1}{\cos \mathrm{x}} & \Rightarrow \frac{\mathrm{d} \mathrm{y}}{\mathrm{d} \mathrm{x}}=\frac{\cos \mathrm{x}(\cos \mathrm{x})+(\sin \mathrm{x}+1) \sin \mathrm{x}}{\cos ^{2} \mathrm{x}} & \quad=\frac{\cos ^{2} \mathrm{x}+\sin ^{2} \mathrm{x}+\sin \mathrm{x}}{\cos ^{2} \mathrm{x}}=\frac{1+\sin \mathrm{x}}{1-\sin ^{2} \mathrm{x}}=\frac{1}{1-\sin \mathrm{x}} & \Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}=\frac{(1-\sin \mathrm{x}) \cdot 0-1(0-\cos \mathrm{x})}{(1-\sin \mathrm{x})^{2}}=\frac{\cos \mathrm{x}}{(1-\sin \mathrm{x})^{2}} \end{aligned} $$