Differentiate the following functions with respect to $x$ :
$\cos ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$
$y=\cos ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$
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
$\frac{d y}{d x}=\frac{d}{d x}\left(\cot ^{-1}\left(\frac{x}{a}\right)\right)$
$\frac{d y}{d x}=\frac{-a^{2}}{a^{2}+x^{2}} \times \frac{1}{a}$
$\frac{d y}{d x}=\frac{-a}{a^{2}+x^{2}}$
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