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gizmo_the_mogwai [7]
3 years ago
12

(E) 10,080

SAT
1 answer:
pogonyaev3 years ago
5 0

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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