Final answer:
To find the instantaneous rate of change of the function f(x) = x^(2/3) at x = 8, calculate the derivative to get f'(x) = (2/3)x^(-1/3) and evaluate it at x = 8 to find that the rate is 1/3.
Step-by-step explanation:
The question deals with finding the instantaneous rate of change of the function f(x) = x2/3 at a specific point, which is x = 8. To find this rate of change, we need to calculate the derivative of the function and then substitute the value x = 8 into the derivative function. The derivative of f(x) = x2/3 using the power rule is f'(x) = (2/3)x-1/3. Substituting x = 8 into the derivative gives us f'(8) = (2/3)8-1/3 = (2/3)(1/2), which simplifies to 1/3. Therefore, the instantaneous rate of change of the function at x = 8 is 1/3.