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!