Differentiate the following functions with respect to x :

$y=\tan ^{-1}\left\{\frac{4 x}{1-4 x^{2}}\right\}$

Let $2 x=\tan \theta$

$y=\tan ^{-1}\left\{\frac{2 \tan \theta}{1-\tan ^{2} \theta}\right\}$

Using $\tan 2 \theta=\frac{2 \tan \theta}{1-\tan ^{2} \theta}$

$y=\tan ^{-1}(\tan 2 \theta)$

Considering the limits,

$-\frac{1}{2}

$-1<2 x<1$

$-1<\tan \theta<1$

$-\frac{\pi}{4}<\theta<\frac{\pi}{4}$

$-\frac{\pi}{2}<2 \theta<\frac{\pi}{2}$

Now,

$y=\tan ^{-1}(\tan 2 \theta)$

$y=2 \theta$

$y=2 \tan ^{-1}(2 x)$

Differentiating w.r.t $\mathrm{x}$, we get

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

$\frac{d y}{d x}=2 \times \frac{2}{1+(2 x)^{2}}$

$\frac{d y}{d x}=\frac{4}{1+4 x^{2}}$