(E) 10,080 48. For which of the following functions is
f(x + 3) = f(x) f(3) for all values of x
in the domain of the function?
(A) f(X) = x
(B) f(x) = ex
(C) f(x) = log x
(D) f(x) sin x
(E)f(x) = tan x

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