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Differentiate f(x) = (2x³)/(4x⁵ + 5) with respect to x.

A) -4/(4x⁵ + 5)²
B) 6x/(4x⁵ + 5)²
C) (6x² - 20x)/(4x⁵ + 5)²
D) -6x/(4x⁵ + 5)²

1 Answer

1 vote

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.

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