Answer :
Final answer:
After differentiating the given function f(x) = 3x^5(x^4 - 3), it is determined that none of the provided multiple-choice options match the correct derivative, which is 27x^8 - 45x^5. Since the correct answer is not listed, the student should re-evaluate the options or the derivation process.
None of the option is correct.
Explanation:
The student has asked us to find f^x when the function f(x) = 3x^5(x^4 - 3). To find f^x, we need to differentiate the function f(x) with respect to x.
Let's derive the given function step by step:
Apply the power rule for differentiation to the term 3x^5, which becomes 15x^4.
Then differentiate the polynomial (x^4 - 3), which results in 4x^3.
Now, follow the product rule: the derivative of two multiplied functions u and v (where u = 3x^5 and v = x^4 - 3) is u'v + uv'.
Substitute the differentiated parts into the product rule formula: (15x^4)(x^4 - 3) + (3x^5)(4x^3).
Simplify the expression: 15x^8 - 45x^5 + 12x^8.
Combine like terms: (15x^8 + 12x^8) - 45x^5, which simplifies to 27x^8 - 45x^5.
None of the options (a), (b), (c), or (d) match 27x^8 - 45x^5, which suggests there might be an error either in the student's provided options or my calculation. Hence, I must refrain from choosing any of the provided answers and advise re-evaluating the work.
None of the option is correct.
