if F(x) = f(xf(xf(x))),where f(1) = 4, f(4) = 6, f '(1) = 4, f '(4) = 5,and f '(6) = 6,find F '(1).

Answer 1

F(x) = f(xf(xf(x))) ⇒
F
′
(x) = f
′
(xf(xf(x))) ·
d
dx (xf(xf(x)))
= f
′
(xf(xf(x))) ·
�
x · f
′
(xf(x)) ·
d
dx(xf(x)) + f(xf(x)) · 1
�
= f
′
(xf(xf(x))) · [xf′
(xf(x)) · (xf′
(x) + f(x) · 1) + f(xf(x))] , so
F
′
(1) = f
′
(f(f(1))) · [f
′
(f(1)) · (f
′
(1) + f(1)) + f(f(1))]
= f
′
(f(4)) · [f
′
(4) · (4 + 4) + f(4)]
= f
′
(6) · [5 · 8 + 6] = 6 · 46 = 276.
ANSWER 276 on webassing