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We are tasked with finding the derivative of the function:

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

Let's compute the derivative \( f'(x) \) step-by-step using the power rule for derivatives, which states:

\[
\frac{d}{dx}(x^n) = n x^{n-1}
\]

### Step 1: Derivative of \( \frac{2}{3}x^{-3} \)

\[
\frac{d}{dx} \left( \frac{2}{3}x^{-3} \right) = \frac{2}{3} \cdot (-3) x^{-4} = -2x^{-4}
\]

### Step 2: Derivative of \( -\frac{1}{2}x^{-4} \)

\[
\frac{d}{dx} \left( -\frac{1}{2}x^{-4} \right) = -\frac{1}{2} \cdot (-4) x^{-5} = 2x^{-5}
\]

### Step 3: Derivative of \( 2x \)

\[
\frac{d}{dx}(2x) = 2
\]

### Step 4: Derivative of \( -\frac{1}{3} \)

Since this is a constant, its derivative is zero.

\[
\frac{d}{dx}\left( -\frac{1}{3} \right) = 0
\]

### Final derivative

Now, combining all the terms together:

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

This matches option **(a)** from the choices.

Would you like further details or have any questions?

Here are five related questions:
1. How would you differentiate \( f(x) = x^{-3} + x^{-4} \)?
2. Can you explain the power rule for differentiation?
3. How do you find the second derivative of a function?
4. What is the derivative of a constant multiplied by a function?
5. How does the derivative behave for exponential functions like \( e^x \)?

**Tip:** Always check if you can simplify expressions before differentiating to make the process easier.

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

Learn how to find the derivative of the function f(x) = (2/3)x^(-3) - (1/2)x^(-4) + 2x - (1/3) using the power rule. A detailed step-by-step explanation is provided, suitable for students in Grades 10-12.

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