Differentiate the following functions with respect to x :

Solution:

$y=\cos ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$

Let $x=a \cot \theta$

Now

$y=\cos ^{-1}\left\{\frac{\operatorname{acot} \theta}{\sqrt{a^{2} \cot ^{2} \theta+a^{2}}}\right\}$

Using $1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$

$y=\cos ^{-1}\left\{\frac{\operatorname{acot} \theta}{a \sqrt{\cot ^{2} \theta+1}}\right\}$

$y=\cos ^{-1}\left\{\frac{\operatorname{acot} \theta}{a \sqrt{\operatorname{cosec}^{2} \theta}}\right\}$

$y=\cos ^{-1}\left\{\frac{\cot \theta}{\operatorname{cosec} \theta}\right\}$

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

$y=\theta$

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

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

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

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

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