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Question
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
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Solution
\[\text{ Case } 1:\]
\[x > 0\]
\[|x| = x\]
\[\text{ Thus, we have }:\]
\[\frac{d}{dx}\left( x|x| \right) = \frac{d}{dx}\left( x . x \right) = \frac{d}{dx}\left( x^2 \right) = 2x \left( 1 \right)\]
\[\text{ Case } 2:\]
\[x < 0\]
\[|x| = - x\]
\[\text{ Thus, we have }:\]
\[\frac{d}{dx}\left( x|x| \right) = \frac{d}{dx}\left( x . \left( - x \right) \right) = \frac{d}{dx}\left( - x^2 \right) = - 2x \left( 2 \right)\]
\[\text{ From } (1) \text{ and } (2), \text{ we have }:\]
\[\frac{d}{dx}\left( x|x| \right) = \binom{2x, if x > 0}{ - 2x, if x < 0}\]
\[\]
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