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.

Find the derivative of the function f(x) = x/(x+5) using the quotient rule. This step-by-step solution explains the quotient rule application, simplifies the result, and highlights key calculus concepts. Suitable for high school and college students studying calculus.

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