Let's find the difference quotient for the function $f(x) = \frac{1}{x+3}$.

The difference quotient is given by:

$\frac{f(x+h) - f(x)}{h}$

1. Calculate $f(x+h)$:

$f(x+h) = \frac{1}{(x+h)+3} = \frac{1}{x+h+3}$

2. Find the difference $f(x+h) - f(x)$:

$f(x+h) - f(x) = \frac{1}{x+h+3} - \frac{1}{x+3}$

To subtract these fractions, we need a common denominator:

$f(x+h) - f(x) = \frac{(x+3) - (x+h+3)}{(x+h+3)(x+3)}$

Simplify the numerator:

$f(x+h) - f(x) = \frac{x + 3 - x - h - 3}{(x+h+3)(x+3)} = \frac{-h}{(x+h+3)(x+3)}$

3. Form the difference quotient:

$\frac{f(x+h) - f(x)}{h} = \frac{-h}{h \cdot (x+h+3)(x+3)}$

Simplify by canceling $h$ in the numerator and denominator:

$\frac{f(x+h) - f(x)}{h} = \frac{-1}{(x+h+3)(x+3)}$

So, the difference quotient for $f(x) = \frac{1}{x+3}$ is:

$\frac{f(x+h) - f(x)}{h} = \frac{-1}{(x+h+3)(x+3)}$

Would you like further clarification or have any questions?

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Here are 5 related questions for further exploration:

1. How does the difference quotient change as $h$ approaches 0 for $f(x) = \frac{1}{x+3}$?
2. What is the derivative of $f(x) = \frac{1}{x+3}$ using the definition of the derivative?
3. How would the difference quotient change for a function with a different denominator, such as $f(x) = \frac{1}{x+5}$?
4. Can you apply the difference quotient to a function like $f(x) = \frac{2}{x+3}$?
5. How does the concept of limits apply to the difference quotient?

Tip: When working with rational functions, always remember to find a common denominator when subtracting fractions. This simplifies the difference quotient and prepares it for taking the limit as $h$ approaches 0.