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(a) Find the first, second, third and fourth order derivatives of the function.
The function is , where is a constant.
Apply derivative on each side with respect to .
The first derivative is .
Again apply derivative on each side with respect to .
The second derivative is .
Again apply derivative on each side with respect to .
The third derivative is .
Again apply derivative on each side with respect to .
The fourth derivative is .
(b)
The second derivative is .
Substitute .
.
(c)
The first derivative is .
The second derivative is .
The third derivative is .
The fourth derivative is .
Observe the odd derivatives and .
General rule for odd order deivatives .
Observe the odd derivatives and .
General rule for even order deivatives .
(a)
The first derivative is .
The second derivative is .
The third derivative is .
The fourth derivative is .
(b) .
(c)
General rule for even order deivatives .
General rule for odd order deivatives .
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