Differential coefficient of ( ^ -1 x) is: - Vidyadip

MCQ

Differential coefficient of $\sec(\tan^{-1}\text{x})$ is:

A
$\frac{\text{x}}{1+\text{x}^2}$
B
$\text{x}\sqrt{1+\text{x}^2}$
C
$\frac{\text{x}}{\sqrt{1+\text{x}^2}}$
✓
$\frac{\text{x}}{\sqrt{1+\text{x}^2}}$

✓

Answer

Correct option: D.

$\frac{\text{x}}{\sqrt{1+\text{x}^2}}$

We have, $\text{y}=\sec(\tan^{-1}\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\sec\big(\tan^{-1}\text{x}\big)\tan\big(\tan^{-1}\text{x}\big)\times\frac{\text{d}}{\text{dx}}\big(\tan^{-1}\text{x}\big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\sec\big(\tan^{-1}\text{x}\big)\tan\big(\tan^{-1}\text{x}\big)\times\frac{1}{\sqrt{1+\text{x}^2}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\Big(\frac{\text{x}}{\sqrt{1+\text{x}^2}}\Big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\Big(\frac{\text{x}}{\sqrt{1+\text{x}^2}}\Big)\text{y}$

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