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Mathematics
Problem
Suppose
f
f
f
and
g
g
g
are differentiable and invertible with
f
(
2
)
=
5
f(2)=5
f
(
2
)
=
5
,
f
′
(
2
)
=
−
4
f'(2)=-4
f
′
(
2
)
=
−
4
,
g
(
3
)
=
5
g(3)=5
g
(
3
)
=
5
, and
g
′
(
3
)
=
2
g'(3)=2
g
′
(
3
)
=
2
. Let
p
(
x
)
=
f
−
1
(
x
)
g
−
1
(
x
)
p(x)=f^{-1}(x)\,g^{-1}(x)
p
(
x
)
=
f
−
1
(
x
)
g
−
1
(
x
)
. Determine
p
′
(
5
)
p'(5)
p
′
(
5
)
. a)
−
3
4
-\dfrac34
−
4
3
b)
−
1
8
-\dfrac18
−
8
1
c)
1
4
\dfrac14
4
1
d)
5
4
\dfrac54
4
5
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