We are given the function:

$F(x) = f(x f(x f(x)))$

and need to determine $F'(1)$.

Step 1: Define $u(x)$

Let:

$u(x) = x f(x f(x))$

so that:

$F(x) = f(u(x))$

Applying the chain rule:

$F'(x) = f'(u(x)) \cdot u'(x).$

Step 2: Differentiate $u(x)$

Define $v(x) = x f(x f(x))$, so:

$u(x) = x f(v(x)).$

Using the product rule:

$u'(x) = f(v(x)) + x f'(v(x)) v'(x).$

Step 3: Differentiate $v(x)$

Since:

$v(x) = x f(x f(x)),$

we differentiate:

$v'(x) = f(x f(x)) + x f'(x f(x)) \cdot (f(x) + x f'(x) f(x)).$

Step 4: Evaluate at $x = 1$

From the given values:

• $f(1) = 4$
• $f(4) = 5$
• $f'(1) = 3$
• $f'(4) = 4$
• $f'(5) = 5$

Compute:

$v(1) = 1 \cdot f(1 f(1)) = 1 \cdot f(4) = 5.$

$v'(1) = f(4) + 1 \cdot f'(4) (f(1) + 1 \cdot f'(1) f(1))$

$= 5 + 4(4 + 3 \times 4) = 5 + 4(4 + 12) = 5 + 4(16) = 5 + 64 = 69.$

$u(1) = 1 \cdot f(v(1)) = f(5) = 5.$

$u'(1) = f(v(1)) + 1 \cdot f'(v(1)) v'(1).$

$= 5 + f'(5) \cdot 69 = 5 + 5 \times 69 = 5 + 345 = 350.$

$F'(1) = f'(u(1)) \cdot u'(1) = f'(5) \cdot 350.$

Since $f'(5) = 5$:

$F'(1) = 5 \times 350 = 1750.$

Final Answer:

$\boxed{1750}$

Would you like a breakdown of any step?

Here are five related questions to explore further:

1. How does the chain rule apply when differentiating nested functions?
2. How do you compute derivatives when dealing with function compositions?
3. What role does the product rule play in differentiating $u(x)$?
4. Why is knowing specific function values crucial in derivative evaluation?
5. How does implicit differentiation apply to function nesting?

Tip: When dealing with composite functions, carefully define intermediate variables to simplify differentiation.