Q.Differentiate the following w.r.t. $x$: $\cot^{-1}\dfrac{\sqrt{1+\sin(4x/3)}+\sqrt{1-\sin(4x/3)}}{\sqrt{1+\sin(4x/3)}-\sqrt{1-\sin(4x/3)}}$

Inverse trigonometric functions ($\sin^{-1}x,\cos^{-1}x,\tan^{-1}x,\cot^{-1}x,\sec^{-1}x,\cosec^{-1}x$) are multi-valued in general, so a principal branch (a restricted domain and range) is fixed for each before differentiating. Each derivative is found by treating $y=$ (inverse trig function of $x$) as $x=$ (trig function of $y$), differentiating implicitly with respect to $y$, and using a Pythagorean identity to express the result back in terms of $x$. The six standard results are: $\dfrac{d}{dx}\sin^{-1}x=\dfrac{1}{\sqrt{1-x^2}}$, $\dfrac{d}{dx}\cos^{-1}x=-\dfrac{1}{\sqrt{1-x^2}}$, $\dfrac{d}{dx}\tan^{-1}x=\dfrac{1}{1+x^2}$, $\dfrac{d}{dx}\cot^{-1}x=-\dfrac{1}{1+x^2}$, $\dfrac{d}{dx}\sec^{-1}x=\dfrac{1}{x\sqrt{x^2-1}}$ (for $x>1$; sign flips for $x<-1$), and $\dfrac{d}{dx}\cosec^{-1}x=-\dfrac{1}{x\sqrt{x^2-1}}$ (for $x>1$; sign flips for $x<-1$). When the argument is a function $f(x)$ rather than pla …