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).
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
To find F '(1), we need to find the derivative of F(x) with respect to x and then substitute x = 1.
Let's start by finding the derivative of F(x) step by step.
1. Start with the given equation F(x) = f(xf(xf(x))).
2. Take the derivative of both sides of the equation with respect to x using the chain rule for nested functions:
F'(x) = f'(xf(xf(x))) * (xf(xf(x)))' (Using the chain rule)
3. Now we need to find the derivative of xf(xf(x)) with respect to x. Again, we can use the chain rule:
(xf(xf(x)))' = x*f'(xf(x)) * (xf(x))' (Using the chain rule)
4. To simplify further, we can determine the derivative of f(xf(x)) with respect to x:
(xf(x))' = f'(xf(x)) * (xf(x))' (Using the chain rule)
5. Now we have all the components to find F'(x). Substituting the results from steps 3 and 4, we get:
F'(x) = f'(xf(xf(x))) * (x*f'(xf(x)) * (xf(x))')
6. Finally, to find F '(1), substitute x = 1 in the expression derived above:
F'(1) = f'(f(1 * f(1)) * (1*f'(f(1)) * (f(1))')
Now we can substitute the given values to find F'(1):
Given values:
f(1) = 4
f(4) = 6
f '(1) = 4
f '(4) = 5
f '(6) = 6
Using these values, we can evaluate F '(1) using the derived expression in step 6.