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The given function is:

\[
g(x) = \sqrt{5x} - \frac{1}{2x}
\]

To find the derivative \( g'(x) \), we will use the following rules of differentiation:

1. The derivative of \( \sqrt{x} = x^{1/2} \) is given by \( \frac{1}{2\sqrt{x}} \).
2. The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).
3. The power rule for derivatives, \( \frac{d}{dx}[x^n] = n x^{n-1} \).

Now, let's compute the derivative of each term.

### Step 1: Derivative of \( \sqrt{5x} \)
We can rewrite \( \sqrt{5x} \) as \( (5x)^{1/2} \). By applying the chain rule:
\[
\frac{d}{dx}\left( (5x)^{1/2} \right) = \frac{1}{2}(5x)^{-1/2} \cdot 5 = \frac{5}{2\sqrt{5x}}
\]

### Step 2: Derivative of \( -\frac{1}{2x} \)
This is \( -\frac{1}{2} \cdot \frac{1}{x} \). Using the rule for \( \frac{1}{x} \), we get:
\[
\frac{d}{dx}\left( -\frac{1}{2x} \right) = \frac{1}{2x^2}
\]

### Final derivative \( g'(x) \)
Putting it all together:
\[
g'(x) = \frac{5}{2\sqrt{5x}} + \frac{1}{2x^2}
\]

This is the derivative of the given function.

Would you like any further explanation or details?

### Related Questions:
1. How can the chain rule be applied to other functions?
2. What is the second derivative of this function \( g(x) \)?
3. How would you compute the derivative of a product of two functions?
4. How do you differentiate a function with multiple variables?
5. Can you explain the process of implicit differentiation?

### Tip:
Always simplify functions as much as possible before differentiating to avoid unnecessary complexity.

Find the derivative of the function g(x) = sqrt(5x) - 1/(2x). This detailed step-by-step solution covers key calculus concepts such as differentiation, chain rule, and power rule. Suitable for university-level students.

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