Differentiate the following functions with respect to x: ^ -1 x x… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\cos^{-1}\Big\{\frac{\text{x}}{\sqrt{\text{x}^2+\text{a}^2}}\Big\}$

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Answer

Let $\text{y}=\cos^{-1}\Big\{\frac{\text{x}}{\sqrt{\text{x}^2+\text{a}^2}}\Big\}$
Let $\text{x}=\text{a}\cot\theta$
$\Rightarrow\ \text{y}=\cos^{-1}\Big\{\frac{\text{a}\cot\theta}{\sqrt{\text{a}^2\cot^2\theta+\text{a}^2}}\Big\}$
$\Rightarrow\text{y}=\cos^{-1}\Big\{\frac{\text{a}\cot\theta}{\sqrt{\text{a}^2(\cot^2\theta+1)}}\Big\}$
$\Rightarrow\ \text{y}=\sin^{-1}\Big(\frac{\text{a}\cot\theta}{\text{a cosec}\theta}\Big)$
$\Rightarrow\ \text{y}=\cos^{-1}\Bigg(\frac{\frac{\cos\theta}{\sin\theta}}{\frac{1}{\sin\theta}}\Bigg)$
$\Rightarrow\ \text{y}=\cos^{-1}(\cos\theta)$
$\Rightarrow\ \text{y}=\theta$
$\Rightarrow\ \text{y}=\cot^{-1}\big(\frac{\text{x}}{\text{a}}\big)\ \big[\text{Since, x}=\text{a}\cot\theta\big]$
Differentiating it with respect to x using chain rule,
$\frac{\text{dy}}{\text{dx}}=\frac{1}{1+\big(\frac{\text{x}}{\text{a}}\big)^2}\frac{\text{d}}{\text{dx}}\big(\frac{\text{x}}{\text{a}}\big)$
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{-\text{a}^2}{\text{a}^2+\text{x}^2}\times\big(\frac{1}{\text{a}}\big)$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{-\text{a}}{\text{a}^2+\text{x}^2}$

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