Differentiate the following functions with respect to x :

Question:

Differentiate the following functions with respect to $x$ :

$\sin ^{-1}\left\{\frac{x}{\sqrt{x^{2}+a^{2}}}\right\}$

Solution:

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

Let $x=a \tan \theta$

Now

$y=\sin ^{-1}\left\{\frac{\operatorname{atan} \theta}{\sqrt{a^{2} \tan ^{2} \theta+a^{2}}}\right\}$

Using $1+\tan ^{2} \theta=\sec ^{2} \theta$

$y=\sin ^{-1}\left\{\frac{a \tan \theta}{a \sqrt{\tan ^{2} \theta+1}}\right\}$

$y=\sin ^{-1}\left\{\frac{\operatorname{atan} \theta}{a \sqrt{\sec ^{2} \theta}}\right\}$

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

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

$y=\theta$

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

Differentiating w.r.t $x$, we get

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

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

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

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