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Answer :
Final answer:
The derivative f'(x) of the function f(x) = x^(2/3) - 4x^9 is found using the power rule and is (2/3)x^(-1/3) - 36x^8, which corresponds to choice (a).
Explanation:
The student has asked for the derivative f'(x) of the function f(x) = x2/3 - 4x9. To find the derivative, we apply the power rule which states that if f(x) = xn, then f'(x) = nxn-1.
Following this rule, the derivative of x2/3 is (2/3)x(2/3)-1 or (2/3)x-1/3. The derivative of -4x9 is -36x8. Combining these gives us:
f'(x) = (2/3)x-1/3 - 36x8
Thus, the correct answer is (a). (2/3)x-1/3 - 36x8.
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