$\"\"$

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The function is $\"\"$ .

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Definition of absolute value of the function $\"\"$ :

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If $\"\"$ , then $\"\"$ and if $\"\"$ , then $\"\"$ .

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Let $\"\"$ ==> $\"\"$ .

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Determine the left hand and right hand limit at $\"\"$ .

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Right hand limit:

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$\"\"$

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Left hand limit:

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$\"\"$

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Since the left hand limit is not equal to right hand limit.

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Therefore $\"\"$ does not exist.

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$\"\"$

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$\"\"$ .

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Write the above absolute value function in piece wise function form and it is defined as

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$\"\"$

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Derivative of the function $\"\"$ is $\"\"$ .

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For $\"\"$ ,

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$\"\"$

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Therefore, $\"\"$ is differentiable for $\"\"$ that is $\"\"$ .

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For $\"\"$ ,

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$\"\"$

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Therefore, $\"\"$ is differentiable for $\"\"$ that is $\"\"$ .

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We write the formula for $\"\"$ as follows.

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$\"\"$

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$\"\"$

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Graph:

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Graph the piecewise function $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .