High School

Use the Product Rule to differentiate \( f(x) = (4x^2 - 4x + 6)(3x^2 + 8x - 2) \).

A. \( f'(x) = 50x^3 + 60x^2 - 44x + 56 \)
B. \( f'(x) = 50x^3 + 60x^2 - 40x + 56 \)
C. \( f'(x) = 48x^3 + 60x^2 - 44x + 56 \)
D. \( f'(x) = 48x^3 + 62x^2 - 44x + 56 \)
E. \( f'(x) = 48x^3 + 60x^2 - 40x + 54 \)

Answer :

Final answer:

To differentiate the function f(x) using the Product Rule, first, differentiate each function separately. Then, according to the Product Rule, multiply the derivative of the first function by the second function and add that to the product of the first function and the derivative of the second function.

Explanation:

The Product Rule in calculus is applied when differentiating the product of two or more functions. Here we have to differentiate the function f (x) = (4x² - 4x + 6). (3x² + 8x - 2). According to the product rule, the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times derivative of the second function.

The differentiation can be performed as follows:

  1. First, differentiate each function separately, we get derivative of (4x² - 4x + 6) is (8x - 4), and derivative of (3x² + 8x - 2) is (6x + 8).
  2. Then apply product rule: f '(x) = (first function's derivative).(second function) + (first function).(second function's derivative)
  3. We get f'(x) = (8x - 4).(3x² + 8x - 2) + (4x² - 4x + 6).(6x + 8)
  4. After the multiplication and addition, we get the final derivative f'(x).

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