$\"\"$

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(a) Find the first, second, third and fourth order derivatives of the function.

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The function is $\"\"$ , where $\"\"$ is a constant.

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

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$\"\"$

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$\"\"$

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$\"\"$

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The first derivative is $\"\"$ .

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$\"\"$

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$\"\"$

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Again apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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The second derivative is $\"\"$ . $\"\"$

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$\"\"$

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Again apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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The third derivative is $\"\"$ .

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$\"\"$

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$\"\"$

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Again apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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The fourth derivative is $\"\"$ .

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$\"\"$

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(b)

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The second derivative is $\"\"$ .

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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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(c)

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The first derivative is $\"\"$ .

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The second derivative is $\"\"$ .

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The third derivative is $\"\"$ .

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The fourth derivative is $\"\"$ .

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Observe the odd derivatives $\"\"$ and $\"\"$ .

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General rule for odd order deivatives $\"\"$ .

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Observe the odd derivatives $\"\"$ and $\"\"$ .

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General rule for even order deivatives $\"\"$ .

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$\"\"$

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(a)

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The first derivative is $\"\"$ .

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The second derivative is $\"\"$ .

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The third derivative is $\"\"$ .

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The fourth derivative is $\"\"$ .

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(b) $\"\"$ .

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(c)

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General rule for even order deivatives $\"\"$ .

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General rule for odd order deivatives $\"\"$ .