https://file.aiquickdraw.com/mathbot/file/1724653880980uzm7va5j.jpg

but in my text book the simplification is like this 

The expression you've provided is:

\[
\frac{x+1}{2x\sqrt{x}}
\]

This can indeed be simplified. Let's work through the steps to see how it simplifies.

### Step 1: Distribute the denominator
First, we can break up the fraction by distributing the denominator to each term in the numerator:

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

### Step 2: Simplify each term

- **First Term:**

\[
\frac{x}{2x\sqrt{x}} = \frac{1}{2\sqrt{x}}
\]

- **Second Term:**

The second term is already in its simplest form:

\[
\frac{1}{2x\sqrt{x}}
\]

### Step 3: Combine the simplified terms

Now, putting it all together:

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

This is the simplified form that was shown in the original differential equation:

\[
\frac{dy}{dx} = \frac{1}{2\sqrt{x}} + \frac{1}{2x\sqrt{x}}
\]

It turns out the expression in your textbook is exactly the same as the simplified form of the original equation you provided earlier! The expression you showed in the book corresponds directly to the derivative expression before it was simplified.

---

Would you like any further details on this, or have any more questions?

### 5 Relative Questions:
1. Why does distributing the denominator work in simplifying expressions?
2. Can you apply this technique to other types of fractions with different denominators?
3. How would this method change if the numerator was a more complex polynomial?
4. What are the implications of such a simplification when solving integrals?
5. Can this process be reversed to combine terms into a single fraction?

### Tip:
When simplifying fractions, look for opportunities to break up the numerator or denominator, especially if it helps to identify common factors that can be simplified.

Learn how to simplify the fraction (x+1)/(2x*sqrt(x)) with detailed step-by-step solutions. This problem involves fraction simplification techniques and is suitable for students in Grades 11-12.

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