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Differentiate each function with respect to x. y = 5x⁵ /3x⁵ +3

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Final answer:

To differentiate the function y = 5x⁵ / (3x⁵ + 3), we can use the quotient rule, which states that (f'(x)g(x) - f(x)g'(x)) / (g(x))². Applying this rule, we find the derivative to be 75x⁴ / (3x⁵ + 3)².

Step-by-step explanation:

To differentiate the function y = 5x⁵ / (3x⁵ + 3), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) / g(x), then its derivative is given by (f'(x)g(x) - f(x)g'(x)) / (g(x))². Applying this rule to the given function, we get:

y' = [(5x⁵)'(3x⁵ + 3) - (5x⁵)(3x⁵ + 3)'] / (3x⁵ + 3)²

Simplifying further:

y' = [25x⁴(3x⁵ + 3) - 5x⁵(15x⁴)] / (3x⁵ + 3)²

y' = (75x⁹ + 75x⁴ - 75x⁹) / (3x⁵ + 3)²

y' = 75x⁴ / (3x⁵ + 3)²

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