Differentiate the following w.r.t $x$ : $\cos\bigg(\tan\sqrt {x+1}\bigg)$.

Question

$\begin{array}{1 1} (A)\;\large\frac{-\sin(\tan\sqrt{x+1}\times \sec^2\sqrt{x+1}}{2\sqrt{x+1}} \\ (B)\;\large\frac{\sin(\tan\sqrt{x+1}\times \sec^2\sqrt{x+1}}{2\sqrt{x+1}} \\ (C)\;\large\frac{-\cos(\tan\sqrt{x+1}\times \sec^2\sqrt{x+1}}{2\sqrt{x+1}} \\ (D)\;\large\frac{\cos(\tan\sqrt{x+1}\times \sec^2\sqrt{x+1}}{2\sqrt{x+1}} \end{array} $