Final answer:
The correct derivative of the function f(x) = (cos⁴(3x) + 7)³ is -36(cos⁴(3x)+7)²cos³(3x)sin(3x), applying the chain rule and power rule for differentiation. None of the options provided in the question are correct.
Step-by-step explanation:
The student is asking to find the derivative of the function f(x) = (cos⁴(3x) + 7)³. To do this, we use the chain rule for differentiation. First, let's differentiate the outer function which is raised to the power of three. The derivative of u³ with respect to u is 3u². So, the derivative of the outer function using u = cos⁴(3x) + 7 is 3(cos⁴(3x) + 7)².
Next, we differentiate the inner function cos⁴(3x), which is a composite function itself. To differentiate cos⁴(3x), we apply the chain rule again, differentiating cos(3x) and raising it to the fourth power. The derivative of cos(3x) with respect to x is -sin(3x) multiplied by the derivative of 3x with respect to x, which is 3. Then, applying the power rule, the derivative of u⁴ with respect to u is 4u³, so the derivative of cos⁴(3x) is 4cos³(3x)(-sin(3x)) ⋅ 3.
Now, we multiply the derivative of the outer function by the derivative of the inner function to get the overall derivative. Therefore, the derivative of f(x) is 3(cos⁴(3x)+7)² ⋅ 4cos³(3x)(-sin(3x)) ⋅ 3. This simplifies to 3 ⋅ 4 ⋅ 3(cos⁴(3x)+7)²cos³(3x)(-sin(3x)), which is -36(cos⁴(3x)+7)²cos³(3x)sin(3x).
Therefore, the correct answer is none of the options provided, because the correct derivative should include the term -36cos³(3x)sin(3x).