Given the function \( f(x) = \frac{x}{x+2} \), we want to find its derivative \( f'(x) \).

### Step 1: Apply the Quotient Rule
The quotient rule is used when we have a function in the form \( \frac{g(x)}{h(x)} \). The derivative is given by:

\[
\frac{d}{dx} \left( \frac{g(x)}{h(x)} \right) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]

For \( f(x) = \frac{x}{x+2} \), we can set:
- \( g(x) = x \)
- \( h(x) = x+2 \)

### Step 2: Differentiate \( g(x) \) and \( h(x) \)
- \( g'(x) = \frac{d}{dx} (x) = 1 \)
- \( h'(x) = \frac{d}{dx} (x+2) = 1 \)

### Step 3: Apply the Quotient Rule
Now, applying the quotient rule:

\[
f'(x) = \frac{1 \cdot (x+2) - x \cdot 1}{(x+2)^2}
\]

Simplifying the numerator:

\[
f'(x) = \frac{x+2 - x}{(x+2)^2} = \frac{2}{(x+2)^2}
\]

Thus, the derivative of \( f(x) \) is:

\[
f'(x) = \frac{2}{(x+2)^2}
\]

### Do you want more details or have any questions?

Here are 5 related questions for further practice:
1. What is the second derivative of \( f(x) = \frac{x}{x+2} \)?
2. Find the derivative of \( f(x) = \frac{x^2}{x+3} \) using the quotient rule.
3. What is the derivative of \( f(x) = \frac{1}{x+2} \)?
4. How do you apply the product rule and quotient rule together?
5. Can you differentiate \( f(x) = \frac{x}{x^2+1} \)?

**Tip:** Always remember to simplify the numerator after applying the quotient rule to avoid calculation errors!

Learn how to find the derivative of the function f(x) = x/(x+2) using the Quotient Rule. This problem provides a detailed step-by-step solution, ideal for students in Grades 11-12 or Calculus 1. It explains key concepts of differentiation and the application of the quotient rule.

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