Question

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

Answer 1

To find f'(1), we differentiate the given equation using the chain rule and substitute the given values. The result is f'(1) = 125.

Step-by-step explanation:

Given the information that f(1) = 4 and f'(1) = 4, we need to find f'(1).

Let's use the chain rule to differentiate the given equation.

1. Let's differentiate both sides of the equation with respect to x.
2. Using the chain rule, we get: f'(x) = f'(xf(xf(x))) * (1 + xf(xf(x)) * f''(xf(xf(x))))
3. Now, we substitute x = 1 into the equation and use the given information: f'(1) = f'(1 * f(1 * f(1))) * (1 + 1 * f(1 * f(1)) * f''(1 * f(1 * f(1))))
4. Substituting f(1) = 4, we simplify the equation: f'(1) = f'(4 * 4 * 4) * (1 + 1 * 4 * f''(4 * 4 * 4))
5. Using the values of f'(4) = 5 and f''(4 * 4 * 4) = 6, we can calculate: f'(1) = 5 * (1 + 1 * 4 * 6) = 5 * (1 + 24) = 5 * 25 = 125
6. Therefore, f'(1) = 125.