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).
Let's take this a step at a time
F' = f'(xf(xf(x))) * (xf(xf(x)))'
(xf(xf(x)))' = f(xf(x)) + xf'(xf(x))
f'(xf(x)) = f(x) + xf'(x)
so:
F'(1) = f'(1f(1f(1))) * (f(1f(1)) + f(1)+f'(1))
= f'(f(4)) * (f(4)+f(1)+f'(1))
= f'(6) (6+4+4)
= 6*14
= 94
Better double-check it, since I don't seem to have used f'(4)=5
94 is not right
To find F'(1), we need to differentiate F(x) with respect to x and then substitute x = 1.
First, let's differentiate F(x) step by step:
1. Apply the chain rule to differentiate the composite function inside F(x):
F'(x) = f'(xf(xf(x))) * (xf(xf(x)))'
2. Next, differentiate xf(xf(x)) using the product rule:
(xf(xf(x)))' = x * f'(xf(x)) * (xf(x))' + f(xf(x)) * (xf(x))'
3. Now, differentiate xf(xf(x)) and xf(x) using the product rule again:
(xf(xf(x)))' = x * [f'(xf(x)) * [x * f'(x) * f(x) + f(x) * f'(x)] + f(xf(x)) * f(x)] + f(xf(x)) * [x * f'(x) + f(x)]
4. Simplify and substitute the given values to find F'(x):
F'(x) = x * [f'(xf(x)) * [x * f'(x) * f(x) + f(x) * f'(x)] + f(xf(x)) * f(x)] + f(xf(x)) * [x * f'(x) + f(x)]
Now, we need to find F'(1). Substituting x = 1 in the expression for F'(x), we have:
F'(1) = 1 * [f'(1f(1)) * [1 * f'(1) * f(1) + f(1) * f'(1)] + f(1f(1)) * f(1)] + f(1f(1)) * [1 * f'(1) + f(1)]
Given that f(1) = 4 and f'(1) = 4, substitute these values into the equation:
F'(1) = 1 * [f'(1f(1)) * [1 * 4 * 4 + 4 * 4] + f(1f(1)) * 4] + f(1f(1)) * [1 * 4 + 4]
Now, substitute the values for f(4), f'(4), and f'(6). However, we need to know the value of f(24) in order to accurately calculate F'(1). Please provide the value of f(24) to continue the calculation.