To differentiate $f(x) = \frac{x}{x+1}$, we can use the quotient rule for differentiation. The quotient rule states that if $f(x) = \frac{g(x)}{h(x)}$, then:

$f'(x) = \frac{g'(x) h(x) - g(x) h'(x)}{[h(x)]^2}$

Here, $g(x) = x$ and $h(x) = x + 1$.

1. Differentiate the numerator $g(x) = x$: $g'(x) = 1$

2. Differentiate the denominator $h(x) = x + 1$: $h'(x) = 1$

Now, applying the quotient rule: $f'(x) = \frac{(1)(x+1) - (x)(1)}{(x+1)^2}$

Simplify the expression: $f'(x) = \frac{x + 1 - x}{(x+1)^2} = \frac{1}{(x+1)^2}$

So, the derivative of $f(x) = \frac{x}{x+1}$ is:

$f'(x) = \frac{1}{(x+1)^2}$

Would you like a detailed explanation of how the quotient rule works?

Here are some related questions:

1. How do we apply the chain rule in combination with the quotient rule?
2. What are common mistakes when differentiating rational functions?
3. Can you explain the product rule and how it differs from the quotient rule?
4. How do derivatives of rational functions behave at vertical asymptotes?
5. What happens to the derivative of this function as $x$ approaches infinity?

Tip: Always check if you can simplify a function before differentiating—it often makes the process easier!