Match the functions in List-I with their deri | vedclass

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(D) The derivatives of the given inverse functions are as follows:$(A)$ $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$ for $|x| > 1$. This ma

Vedclass · Accepted Answer

$A-III, B-I, C-IV, D-II$

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(D) The derivatives of the given inverse functions are as follows:$(A)$ $\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}$ for $|x| > 1$. This matches $III$.$(B)$ $\frac{d}{dx}(	anh^{-1} x) = \frac{1}{1-x^2}$ for $x \in (-1, 1)$. This matches $I$.$(C)$ $\frac{d}{dx}(\coth^{-1} x) = \frac{1}{1-x^2}$ for $x \in R - [-1, 1]$. This matches $IV$.$(D)$ $\frac{d}{dx}(\operatorname{cosech}^{-1} x) = \frac{-1}{|x|\sqrt{x^2+1}}$ for $x 
eq 0$. This matches $II$.Thus, the correct matching is $A-III, B-I, C-IV, D-II$.

List-$I$	List-$II$
$A$. $\sec^{-1} x$	$I$. $\frac{1}{1-x^2}, x \in (-1, 1)$
$B$. $\tanh^{-1} x$	$II$. $\frac{-1}{\|x\| \sqrt{x^2+1}}, x \neq 0$
$C$. $\coth^{-1} x$	$III$. $\frac{1}{\|x\| \sqrt{x^2-1}}, \|x\| > 1$
$D$. $\operatorname{cosech}^{-1} x$	$IV$. $\frac{1}{1-x^2}, x \in R - [-1, 1]$
	$V$. $\frac{-1}{\|x\| \sqrt{1-x^2}}, \|x\| < 1, x \neq 0$

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