Question

Differentiate ${\tan }^{-1}\left(\frac{3a^{2}x-x^3}{a^3-3a{x}^2}\right)$, $-\frac{1}{\sqrt{3}}<\frac{x}{a}<\frac{1}{\sqrt{3}}$

Answer 1

$f\left(x\right)={\tan }^{-1}\left(\frac{3a^{2}x-x^3}{a^3-3a{x}^2}\right)$

$={\tan }^{-1}\left(\frac{3\frac{x}{a}-{\left(\frac{x}{a}\right)}^3}{1-3{\left(\frac{x}{a}\right)}^2}\right)$

Let $\frac{x}{a}=\tan \theta$

$={\tan }^{-1}\left(\frac{3\tan \theta -{\tan }^3\theta }{1-3{\tan }^2\theta }\right)$

$={\tan }^{-1}\left(\tan \right(3\theta \left)\right)$

$=3\theta$

$f\left(x\right)=3{\tan }^{-1}\left(\frac{x}{a}\right)$

$\frac{d}{dx}f\left(x\right)=3\frac{d}{dx}\left({\tan }^{-1}\left(\frac{x}{a}\right)\right)$

$=\frac{3}{1+{\left(\frac{x}{a}\right)}^2}\frac{1}{a}$

$=\frac{3a}{a^2+x^2}$.