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To solve the derivative of the given function, we first need to identify the function and then apply the appropriate differentiation rules.

The function given is:

\[
z = \frac{4x + 2}{2x}
\]

### Step 1: Simplify the Function
First, let's simplify the function before differentiating it.

\[
z = \frac{4x}{2x} + \frac{2}{2x} = 2 + \frac{1}{x}
\]

So, the function simplifies to:

\[
z = 2 + \frac{1}{x}
\]

### Step 2: Differentiate the Function
Now, let's differentiate the simplified function with respect to \( x \).

The derivative of \( z = 2 + \frac{1}{x} \) is:

\[
\frac{dz}{dx} = \frac{d}{dx}(2) + \frac{d}{dx}\left(\frac{1}{x}\right)
\]

- The derivative of a constant \( 2 \) is \( 0 \).
- The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).

So:

\[
\frac{dz}{dx} = 0 - \frac{1}{x^2} = -\frac{1}{x^2}
\]

### Final Answer
The derivative of the function \( z = \frac{4x + 2}{2x} \) with respect to \( x \) is:

\[
\frac{dz}{dx} = -\frac{1}{x^2}
\]

Would you like more details, or do you have any questions? 

Here are some related questions you might be interested in:
1. What is the derivative of a function involving natural logarithms, like \( \ln(x) \)?
2. How can you find the derivative of a product of functions using the product rule?
3. How would the derivative change if the function had been a quotient of two polynomials?
4. What are higher-order derivatives, and how are they calculated?
5. How do you apply the chain rule to differentiate composite functions?

**Tip:** When simplifying functions before differentiating, always check if terms can be simplified to avoid applying complex rules unnecessarily.

Learn how to solve the derivative of the function z = (4x + 2) / 2x, involving algebraic simplification and differentiation. This step-by-step guide breaks down the rules of differentiation and simplification. Ideal for students in Grades 10-12 studying calculus.

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