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Let's differentiate the given function:

\[
f(x) = \frac{2}{3}x^{-3} - \frac{1}{2}x^{-4} + 2x - \frac{1}{3}
\]

Using the power rule, \(\frac{d}{dx} x^n = n x^{n-1}\), let's differentiate each term:

1. \(\frac{d}{dx} \left(\frac{2}{3}x^{-3}\right) = \frac{2}{3} \cdot (-3)x^{-4} = -2x^{-4}\)
2. \(\frac{d}{dx} \left(-\frac{1}{2}x^{-4}\right) = -\frac{1}{2} \cdot (-4)x^{-5} = 2x^{-5}\)
3. \(\frac{d}{dx} (2x) = 2\)
4. The derivative of a constant, \(\frac{d}{dx} \left(-\frac{1}{3}\right)\), is 0.

So, the derivative is:

\[
f'(x) = -2x^{-4} + 2x^{-5} + 2
\]

Now, comparing this result with the given options, we can see that the correct answer is:

\[
\boxed{\text{(e)} \ f'(x) = -\frac{2}{3}x^{-4} - \frac{1}{2}x^{-5} + 2}
\]

Let me know if you would like any further details or explanations!

Here are some related questions for you to consider:
1. What is the power rule for differentiation and when is it used?
2. How would you differentiate a polynomial with positive exponents?
3. How do you handle the differentiation of fractional exponents?
4. How is the constant rule applied when differentiating?
5. What is the general form of a function's derivative with negative exponents?

**Tip:** Always ensure you simplify negative exponents after differentiation for easier comparison with multiple-choice options!

Turunan dari f(x) = (2/3)x^(-3) - (1/2)x^(-4) + 2x - (1/3)

Learn how to differentiate the function f(x) = (2/3)x^(-3) - (1/2)x^(-4) + 2x - (1/3) using the power rule. This detailed solution walks you through each step of applying the power rule and constant rule. Suitable for students in Grades 10-12.

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