$\"\"$

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The magic potential is $\"\"$ .

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Where $\"\"$ and $\"\"$ are constants.

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Find the value of magic potential.

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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Substitute $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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If $\"\"$ then $\"\"$ .

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Substitute the corresponding values in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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If $\"\"$ then $\"\"$ .

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Apply trigonometric identity : $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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Therefore, the magic potential is $\"\"$ .

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$\"\"$

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The magic potential is $\"\"$ .