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The question asks for the derivative of the function \( f(x) = \frac{1}{x^5} + \frac{6}{x^{3/2}} - \frac{2}{3x} \).

To solve this, let's differentiate each term of the function \( f(x) \):

1. **First term: \( \frac{1}{x^5} \)**  
   Rewrite as \( x^{-5} \).  
   The derivative is \( \frac{d}{dx} \left(x^{-5}\right) = -5x^{-6} = -\frac{5}{x^6} \).

2. **Second term: \( \frac{6}{x^{3/2}} \)**  
   Rewrite as \( 6x^{-3/2} \).  
   The derivative is \( \frac{d}{dx} \left(6x^{-3/2}\right) = -9x^{-5/2} = -\frac{9}{x^{5/2}} \).

3. **Third term: \( -\frac{2}{3x} \)**  
   Rewrite as \( -\frac{2}{3} \cdot x^{-1} \).  
   The derivative is \( \frac{d}{dx} \left(-\frac{2}{3}x^{-1}\right) = \frac{2}{3}x^{-2} = \frac{2}{3x^2} \).

Now, putting all the derivatives together, we have:

\[
f'(x) = -\frac{5}{x^6} - \frac{9}{x^{5/2}} + \frac{2}{3x^2}
\]

Looking at the options provided in the image, the correct choice would be:

**Option a: \( -\frac{5}{x^6} - \frac{9}{x^{5/2}} + \frac{2}{3x^2} \)**

Would you like any further details on this solution? Here are some related questions:

1. What are the rules for differentiating functions involving powers of \( x \)?
2. How can you simplify complex rational expressions?
3. Can you explain how to differentiate negative powers of \( x \)?
4. What are the applications of derivatives in real-world problems?
5. How does the chain rule apply to more complex functions?

**Tip:** When differentiating functions, always simplify the expression first for easier differentiation.

Find the derivative of f(x) = 1/x^5 + 6/x^(3/2) - 2/(3x).

Learn how to differentiate the function f(x) = 1/x^5 + 6/x^(3/2) - 2/(3x) using the power rule. This problem involves basic calculus concepts, including differentiation rules for powers of x, and is suitable for advanced high school or early university students.

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