Find the derivative of the function \(\frac{30x^{11} - 45x^6 - 10}{5x^2}\). Simplify your answer without negative exponents.

Which of the following options represents the simplified derivative?

A) \(\frac{66x^{10} - 18x^5}{5x^2}\)

B) \(\frac{66x^{10} - 18x^5}{5x^3}\)

C) \(\frac{66x^{11} - 18x^6}{5x^2}\)

D) \(\frac{66x^{11} - 18x^6}{5x^3}\)

To find the derivative of the function, we apply the quotient rule, which states that the derivative of a quotient of two functions is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

Simplifying the derivative expression by applying the power rule and basic algebraic manipulations yields the final result. In this case, we identify the correct answer by comparing the simplified derivative expression with the options provided. The correct option matches the expression (66[tex]x^{10[/tex]- 18[tex]x^5[/tex])/(5x²). This process confirms our final answer choice.

The quotient rule involves differentiating the numerator and the denominator separately, then combining them using algebraic operations. By applying this rule and simplifying the expression, we obtain the derivative in a form without negative exponents. Comparing this simplified expression with the given options allows us to select the correct answer.

The correct choice matches the expression (66[tex]x^{10[/tex]- 18[tex]x^5[/tex])/(5x²), ensuring that we have correctly found the derivative of the function. This methodical approach ensures accuracy in determining the correct derivative and selecting the appropriate option among the given choices.

So, the correct option is a).