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

Find the derivative of the function \( F(x) = \sqrt[4]{4 + 3x + x^3} \).

Express the function in power form.

The function can be rewritten as \( F(x) = (4 + 3x + x^3)^{\frac{1}{4}} \).

Apply the chain rule to find the derivative.

Using the chain rule, the derivative is given by \( F'(x) = \frac{1}{4}(4 + 3x + x^3)^{-\frac{3}{4}} \cdot (3x^2 + 3) \).

Simplify the expression for the derivative.

The simplified form of the derivative is \( F'(x) = \frac{0.75x^2 + 0.75}{(x^3 + 3x + 4)^{\frac{3}{4}}} \).

The derivative is \( F'(x) = \frac{0.75x^2 + 0.75}{(x^3 + 3x + 4)^{\frac{3}{4}}} \).

The final answer is \( \boxed{F'(x) = \frac{0.75x^2 + 0.75}{(x^3 + 3x + 4)^{\frac{3}{4}}}} \).