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.