Final answer:
The derivative of the function F(x) = ∛(3+5x+x³) is found by applying the chain rule. It results in F'(x) = ⅓ × (5+3x²) / ∛(3+5x+x³)² when simplified.
Step-by-step explanation:
To find the derivative of the function F(x) = ∛(3+5x+x³), we need to apply the chain rule for differentiation. The outer function is the cube root and the inner function is the polynomial 3+5x+x³. Applying the chain rule, we first take the derivative of the outer function, which is ⅓∛(u)², where u is the inner function, and then multiply it by the derivative of the inner function (which is 5 + 3x²).
The derivative of F(x) is therefore given by:
F'(x) = ⅓∛(3+5x+x³)² × (5+3x²)
To express this more clearly, we can write it as:
F'(x) = ⅓ × (5+3x²) / ∛(3+5x+x³)²
Simplifying this expression will give us the final derivative of F(x).