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Mathematics
Problem
Let
f
:
(
0
,
∞
)
→
(
−
∞
,
∞
)
f:(0,\infty)\to(-\infty,\infty)
f
:
(
0
,
∞
)
→
(
−
∞
,
∞
)
be defined as
f
(
x
)
=
e
x
+
ln
x
f(x)=e^x+\ln x
f
(
x
)
=
e
x
+
ln
x
and
g
=
f
−
1
g=f^{-1}
g
=
f
−
1
, then (a)
g
′
(
e
)
=
1
−
e
(
1
+
e
)
3
g'(e)=\frac{1-e}{(1+e)^3}
g
′
(
e
)
=
(
1
+
e
)
3
1
−
e
(b)
g
′
′
(
e
)
=
e
−
1
(
1
+
e
)
3
g''(e)=\frac{e-1}{(1+e)^3}
g
′′
(
e
)
=
(
1
+
e
)
3
e
−
1
(c)
g
′
(
e
)
=
e
+
1
g'(e)=e+1
g
′
(
e
)
=
e
+
1
(d)
g
′
(
e
)
=
1
e
+
1
g'(e)=\frac{1}{e+1}
g
′
(
e
)
=
e
+
1
1
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