High School

Find the derivative of: \( f(x) = \frac{7x^5 - 4}{3x^2 - 2} \).

A. \( f'(x) = \frac{63x^6 - 70x^4 + 24x}{(3x^2 - 2)^2} \)

B. \( f'(x) = \frac{-63x^6 + 70x^4 - 24x}{(3x^2 - 2)^2} \)

Answer :

Final answer:

The derivative of f(x) = (7x⁵ - 4) / (3x² - 2) is (63x⁶ - 70x⁴ + 24x) / ((3x² - 2)²).

Explanation:

To find the derivative of f(x) = (7x⁵ - 4) / (3x² - 2), we will use the quotient rule. The quotient rule states that if we have a function in the form of f(x) = g(x) / h(x), where both g(x) and h(x) are differentiable functions, then the derivative of f(x) is given by:

f'(x) = (h(x)g'(x) - g(x)h'(x)) / (h(x))²

Using the quotient rule, we can differentiate f(x) step by step:

  1. Find the derivatives of 7x⁵ - 4 and 3x² - 2.
  2. Plug the derivatives into the quotient rule formula.
  3. Simplify the expression to get the final result.

After simplifying the expression, we obtain the derivative of f(x) as:
f'(x) = (63x⁶ - 70x⁴ + 24x) / ((3x² - 2)²)

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