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Mathematics
Mathematics
Problem
Let
h
(
x
)
=
cot
(
x
)
h(x)=\cot\bigl(\sqrt{x}\,\bigr)
h
(
x
)
=
cot
(
x
)
for
x
>
0
x>0
x
>
0
. a. Determine an explicit expression for
h
′
(
x
)
h'(x)
h
′
(
x
)
in terms of elementary trigonometric functions of
x
\sqrt{x}
x
and powers of
x
x
x
. b. Compute
h
′
′
(
x
)
h''(x)
h
′′
(
x
)
and simplify as far as possible. c. Evaluate
h
′
′
(
π
2
4
)
h''\!\left(\dfrac{\pi^{2}}{4}\right)
h
′′
(
4
π
2
)
exactly.
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