Let's simplify the expression \(\frac{\left(\frac{1}{x} - \frac{1}{x+1}\right)}{\frac{1}{x+1}}\).

### Step 1: Simplify the numerator
The numerator is \(\frac{1}{x} - \frac{1}{x+1}\).

To subtract these fractions, find a common denominator:

\[
\frac{1}{x} - \frac{1}{x+1} = \frac{(x+1) - x}{x(x+1)} = \frac{1}{x(x+1)}
\]

So, the numerator simplifies to \(\frac{1}{x(x+1)}\).

### Step 2: Express the entire fraction
Now, the expression becomes:

\[
\frac{\frac{1}{x(x+1)}}{\frac{1}{x+1}}
\]

### Step 3: Simplify the fraction
To simplify \(\frac{\frac{1}{x(x+1)}}{\frac{1}{x+1}}\), multiply the numerator by the reciprocal of the denominator:

\[
\frac{1}{x(x+1)} \times \frac{x+1}{1} = \frac{1 \times (x+1)}{x(x+1)} = \frac{1}{x}
\]

So, the simplified expression is \(\frac{1}{x}\).

### Final Answer
\[
\frac{\left(\frac{1}{x} - \frac{1}{x+1}\right)}{\frac{1}{x+1}} = \frac{1}{x}
\]

Would you like further details on any step, or do you have additional questions?

Here are 5 related questions for further exploration:

1. How would the expression change if the numerator was \(\frac{1}{x} + \frac{1}{x+1}\) instead?
2. What is the derivative of the simplified expression \(\frac{1}{x}\) with respect to \(x\)?
3. How can this problem be solved using L'Hôpital's rule if \(x\) approaches a critical point?
4. What happens to the original expression as \(x\) approaches infinity?
5. Can you simplify \(\frac{\frac{1}{x} + \frac{1}{x+1}}{\frac{1}{x-1}}\)?

**Tip:** When dealing with complex fractions, always look to simplify the numerator and denominator separately before combining them.

Learn how to simplify the complex fraction ((1/x)-(1/(x+1)))/(1/(x+1)). This problem involves fraction simplification and finding common denominators, suitable for advanced high school students.

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