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Question
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
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Solution
\[\text{ Case } 1: x>0\]
\[\left| x \right| = x . . . \left( 1 \right)\]
\[\frac{d}{dx}\left( \log \left| x \right| \right) = \log x\]
\[ = \frac{1}{x}\]
\[ = \frac{1}{\left| x \right|} (\text{ from } (1))\]
\[Case 2:x<0\]
\[\left| x \right| = - x . . . \left( 2 \right)\]
\[\frac{d}{dx}\left( \log \left| x \right| \right) = \log \left( - x \right)\]
\[ = \frac{1}{- x}\]
\[ = \frac{1}{\left| x \right|} (\text{ from } (2))\]
\[\text{ From case } (1) \text{ and case }(2),\]
\[\frac{d}{dx}\left( \log \left| x \right| \right) = \frac{1}{\left| x \right|}\]
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