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------------------------------------------------ Find the derivative and simplify.

\[ f(x) = (x^{12} + 4x^4 + 3)(5x^3 - 1) \]

A. \( f'(x) = 140x^{15} + 560x^7 \)

B. \( f'(x) = 140x^{15} + 280x^7 \)

C. \( f'(x) = 70x^{15} + 560x^7 \)

D. \( f'(x) = 70x^{15} + 140x^7 \)

The question is a high school level mathematics problem about calculating the derivative of a given function by using the product rule, simplifying it, and finding the correct option from multiple choices.

Explanation:

The subject of the question is Mathematics, and it pertains to finding and simplifying the derivative of a given function. The student is asked to differentiate the function f(x) = (x12 + 4x4 + 3)(5x3 - 1). To do this, we apply the product rule which states that the derivative of a product of two functions is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function.

Applying the product rule to f(x), we get:

f'(x) = (x12 + 4x4 + 3)'(5x3 - 1) + (x12 + 4x4 + 3)(5x3 - 1)'

After calculating the derivatives and simplifying, we obtain the correct derivative. Comparing the result to the multiple-choice options given, one should match with the simplified result.

So, the correct answer is:a) (f'(x) = 140x15 + 560x⁷).