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Question
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Solution
\[\text{ Case } 1: x>0\]
\[\left| x \right| = x\]
\[f\left( x \right) = \frac{x^2}{\left| x \right|} = \frac{x^2}{x} = x\]
\[f'\left( x \right) = 1\]
\[\text{ Case } 2: x<0\]
\[\left| x \right| = - x\]
\[f\left( x \right) = \frac{x^2}{\left| x \right|} = \frac{x^2}{- x} = - x\]
\[f'\left( x \right) = - 1\]
\[\text{ From case 1 and case 2, we have }:\]
`f'(x)={(1, if, x > 0),(-1, if, x < 0):}`
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