Question

Assertion :Let $f\left(x\right)=\log \left|x\right|$ for $x<0$ then $\frac{d}{dx}\left(\log \left|x\right|\right)=\frac{1}{x}$ Reason: $\left|x\right|=\{\begin{array}{cc}x& \text{if}x\ge 0\\ -x& \text{if}x<0\end{array}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)=\log \left|x\right|$
$f\left(x\right)=\log x$ for $x>0$
$\therefore f^{\prime }\left(x\right)=\frac{1}{x}$ for $x>0$
Now for $x<0,f\left(x\right)=\log \left|x\right|=\log (-x)$
$\therefore f^{\prime }\left(x\right)=\frac{d}{dx}\left(\log (-x)\right)=-\frac{1}{x}(-1)=\frac{1}{x}$
Hence, assertion is correct.
Reason:$\left|x\right|=\{\begin{array}{cc}x& \text{if}x\ge 0\\ -x& \text{if}x<0\end{array}$
Reason is correct.
Hence, reason is the correct explanation for assertion