Final Answer:
The derivative of f(y) = 1/y² - 3/y⁴ * (y⁹ - y³) with respect to y is f'(y) = -2/y³ + 18/y⁵ - 12/y⁷.
Step-by-step explanation:
To differentiate the function f(y) = 1/y² - 3/y⁴ * (y⁹ - y³), we'll apply the rules of differentiation step by step. The function consists of two terms: 1/y² and -3/y⁴ * (y⁹ - y³).
Differentiating 1/y² with respect to y involves applying the power rule for derivatives. The derivative of y⁻² is -2/y³. For the second term, we use the product rule, which involves differentiating each part of the term separately and then combining them. First, the derivative of -3/y⁴ is 12/y⁵ (by applying the power rule). Then, using the chain rule, we differentiate (y⁹ - y³) with respect to y, resulting in 9y⁸ - 3y².
Finally, we multiply the derivative of the first part (-3/y⁴) with the derivative of the second part (9y⁸ - 3y²), giving us -3/y⁴ * (9y⁸ - 3y²) = -27y⁸/y⁴ + 9y²/y⁴ = -27/y⁴ + 9/y². Combining the derivatives of both terms, we get the final derivative: f'(y) = -2/y³ + 18/y⁵ - 12/y⁷.