Differentiate the following functions with respect to x :

Solution:

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

Let $\mathrm{a}^{\mathrm{x}}=\tan \theta$

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

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

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

Considering the limits,

$-\infty

$a^{-\infty}

$0<\tan \theta<1$

$0<\theta<\frac{\pi}{4}$

$0<2 \theta<\frac{\pi}{2}$

Now,

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

$y=2 \theta$

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

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

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

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

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