To find the derivative of $f(x) = \frac{x}{x+5}$, we can use the quotient rule. The quotient rule is given by:

$\left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2}$

Here, $u(x) = x$ and $v(x) = x+5$.

Step 1: Find the derivatives of $u(x)$ and $v(x)$:

$u'(x) = \frac{d}{dx}(x) = 1$ $v'(x) = \frac{d}{dx}(x+5) = 1$

Step 2: Apply the quotient rule:

$f'(x) = \frac{(x+5)(1) - x(1)}{(x+5)^2}$ Simplify the numerator: $f'(x) = \frac{x+5 - x}{(x+5)^2}$ $f'(x) = \frac{5}{(x+5)^2}$

So, the derivative of $f(x) = \frac{x}{x+5}$ is: $f'(x) = \frac{5}{(x+5)^2}$

Would you like further details on this process or have any other questions?

Here are 5 related questions:

1. What is the derivative of $\frac{1}{x+5}$?
2. How do you apply the product rule in differentiation?
3. What is the chain rule in calculus and how is it used?
4. Can you find the second derivative of $\frac{x}{x+5}$?
5. How do you find the critical points of a function?

Tip: Always simplify the numerator and denominator separately before applying the quotient rule to avoid mistakes.