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