1 answer:
Answer:
F '( 1 ) = 84
Explanation:
We will differentiate
F
and wade our way slowly and methodically through finding the derivative of the right. First with the outermost function and chain rule:
F
(
x
)
=
f
(
x
f
(
x
f
(
x
)
)
)
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
(
d
d
x
x
f
(
x
f
(
x
)
)
)
Now applying the product rule:
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
(
d
d
x
f
(
x
f
(
x
)
)
)
]
Reapplying the chain rule:
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
(
f
'
(
x
f
(
x
)
)
+
(
d
d
x
x
f
(
x
)
)
)
]
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
f
'
(
x
f
(
x
)
)
+
x
(
d
d
x
x
f
(
x
)
)
]
Product rule once more:
F
'
(
x
)
=
f
'
(
x
f
(
x
f
(
x
)
)
)
[
f
(
x
f
(
x
)
)
+
x
f
'
(
x
f
(
x
)
)
+
x
(
f
(
x
)
+
x
f
'
(
x
)
)
]
We could simplify this a little more, but I'm dubious it would help. Evaluating at
x
=
1
:
F
'
(
1
)
=
f
'
(
f
(
f
(
1
)
)
)
[
f
(
f
(
1
)
)
+
f
'
(
f
(
1
)
)
+
1
(
f
(
1
)
+
f
'
(
1
)
)
]
F
'
(
1
)
=
f
'
(
f
(
2
)
)
[
f
(
2
)
+
f
'
(
2
)
+
2
+
4
]
F
'
(
1
)
=
f
'
(
3
)
(
3
+
5
+
2
+
4
)
F
'
(
1
)
=
6
(
14
)
F
'
(
1
)
=
84
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