Final answer:
The derivative of the function f(x) = (9/x^(2/3) - 4)^(4/3) is found using the chain rule. After taking the derivative of the inner and outer functions and simplifying, we find that the correct derivative is f'(x) = -24/x^(5/3).
Therefore, the correct answer is (a) f'(x) = -24/x^(5/3).
Step-by-step explanation:
To find the derivative of the function f(x) = (9/x^(2/3) - 4)^(4/3), we'll use the chain rule which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
- First, let's denote the inner function by g(x) = 9/x^(2/3) - 4 and the outer function by h(g) = g^(4/3).
- Now, we take the derivative of the outer function with respect to g, h'(g) = (4/3)g^(1/3).
- Next, the derivative of the inner function g(x) with respect to x is g'(x) = - (6/x^(5/3)).
- Applying the chain rule: f'(x) = h'(g(x)) * g'(x).
- Substitute g(x) and g'(x) into h'(g(x)) * g'(x) and simplify to get: f'(x) = (4/3)*(9/x^(2/3) - 4)^(1/3)*(-(6/x^(5/3))).
- Simplifying further, we can factor and cancel terms which gives us the final answer: f'(x) = -24/x^(5/3).
Therefore, the correct answer is (a) f'(x) = -24/x^(5/3).