Consider the function

(a) Find a formula for the
th derivative
of
and then find
(b) Find the Taylor series for
at
i.e. write
in the form

Solution:
(a)
ExpandStep 1:
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We have
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and
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If we compare these three equations, we notice a pattern.
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We have
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ExpandStep 2:
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Since
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we have
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(b)
ExpandStep 1:
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Since
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we have
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Therefore, the coefficients of the Taylor series are
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ExpandStep 2:
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Therefore, the Taylor series for at is
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ExpandFinal Answer:
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(a)
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(b)
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