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Question
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Solution
\[\text{ Case } 1:\]
\[x > 0\]
\[\left| x \right| = x\]
\[\left( x + \left| x \right| \right)\left| x \right|\]
\[ = \left( x + x \right)x\]
\[ = 2 x^2 \]
\[\frac{d}{dx}\left[ \left( x + \left| x \right| \right)\left| x \right| \right] = \frac{d}{dx}\left( 2 x^2 \right) = 4x \left( 1 \right)\]
\[\text{ Case } 2:\]
\[x < 0\]
\[\left| x \right| = - x\]
\[\left( x + \left| x \right| \right)\left| x \right|\]
\[ = \left( x - x \right)x\]
\[ = 0\]
\[\frac{d}{dx}\left[ \left( x + \left| x \right| \right)\left| x \right| \right] = \frac{d}{dx}\left( 0 \right) = 0 \left( 2 \right)\]
\[\text{ From } (1) \text{and} (2), \text{ we have}:\]
\[\frac{d}{dx}\left[ \left( x + \left| x \right| \right)\left| x \right| \right] = \binom{4x, if x > 0}{0, if x < 0}\]
\[\]
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