Final answer:
The differentiation of f(x) = (2x³)/(4x⁵ + 5) using the quotient rule does not yield any of the provided answer choices, suggesting an error in the differentiation or simplification process.
Step-by-step explanation:
The differentiation of the function f(x) = (2x³)/(4x⁵ + 5) with respect to x involves using the quotient rule: If u(x) and v(x) are differentiable functions of x, then the derivative of u/v is given by (u'v - uv')/v².
Firstly, let's differentiate the numerator u(x)=2x³ and the denominator v(x)=4x⁵+5 separately. The derivative of the numerator is u'(x)=6x², and the derivative of the denominator is v'(x)=20x⁴.
Now we apply the quotient rule:
f'(x) = (6x²(4x⁵+5) - 2x³*20x⁴)/(4x⁵+5)²
Simplifying the numerator we get:
(24x⁷ + 30x² - 40x⁷)/(4x⁵+5)² = (-16x⁷ + 30x²)/(4x⁵+5)² = ((-16/4)x⁷ + (30/4)x²)/(x⁵+5/4)²
Further simplification leads us to:
(-4x⁷ + 7.5x²)/(4x⁵+5)². This is not one of the provided answer choices, indicating that there may have been a mistake in the differentiation process or in the simplification steps.