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Write the Value of D D X { ( X + | X | ) | X | } - Mathematics

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Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]

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Solution

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

Concept: The Concept of Derivative - Algebra of Derivative of Functions
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Q 6 | Page 47

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