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The function given is:

\[
F(x) = \sqrt[4]{4 + 2x + x^3}
\]

We need to find the derivative of this function. To do this, we will apply the chain rule.

### Steps:
1. **Rewrite the function** in exponent form:
\[
F(x) = (4 + 2x + x^3)^{1/4}
\]

2. **Apply the chain rule**. The derivative of \( F(x) = g(x)^{n} \) is \( F'(x) = n \cdot g(x)^{n-1} \cdot g'(x) \), where \( g(x) = 4 + 2x + x^3 \).

3. **Differentiate \( g(x) = 4 + 2x + x^3 \)**:
\[
g'(x) = 0 + 2 + 3x^2 = 2 + 3x^2
\]

4. **Differentiate \( F(x) \)**:
\[
F'(x) = \frac{1}{4} \cdot (4 + 2x + x^3)^{-\frac{3}{4}} \cdot (2 + 3x^2)
\]

Thus, the derivative is:

\[
F'(x) = \frac{1}{4} \cdot (4 + 2x + x^3)^{-\frac{3}{4}} \cdot (2 + 3x^2)
\]

Would you like more details or have any questions?

### 5 Related Questions:
1. How would you differentiate the same function if it were a cube root instead of a fourth root?
2. Can you explain why the chain rule is necessary in this problem?
3. How do we handle higher-order derivatives for functions like this?
4. How does the derivative of a function with fractional exponents differ from regular polynomials?
5. What is the importance of the power rule in this derivative calculation?

### Tip:
When dealing with roots, always rewrite them as fractional exponents to simplify differentiation.

Find the derivative of the function F(x) = 4√(4 + 2x + x^3).

Learn how to differentiate the function F(x) = 4√(4 + 2x + x^3) using the chain rule and power rule. This calculus problem provides a step-by-step solution, including rewriting the root as a fractional exponent and applying key differentiation rules. Suitable for high school and early university students studying Calculus.

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