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The problem you are working on involves simplifying expressions and using the difference quotient to compute the derivative of a function.

The function provided is:
\[ f(x) = \frac{1}{3x + 7} \]

### Part 1: Simplifying \( f(x + h) \)
You correctly simplified \( f(x + h) \) as:
\[ f(x + h) = \frac{1}{3(x + h) + 7} = \frac{1}{3x + 3h + 7} \]

### Part 2: Simplifying the Difference Quotient
Next, you are asked to simplify the difference quotient:
\[
\frac{f(x + h) - f(x)}{h}
\]
Where:
\[
f(x + h) = \frac{1}{3x + 3h + 7}, \quad f(x) = \frac{1}{3x + 7}
\]

Substitute the expressions for \( f(x + h) \) and \( f(x) \):
\[
\frac{\frac{1}{3x + 3h + 7} - \frac{1}{3x + 7}}{h}
\]

To simplify this expression:
1. Find a common denominator for \( \frac{1}{3x + 3h + 7} \) and \( \frac{1}{3x + 7} \).
2. Subtract the two fractions.
3. Simplify the result.
4. Divide the difference by \( h \).

Would you like me to guide you step-by-step through the simplification process for this quotient?

### 5 Related Questions:
1. What are the steps to find a common denominator between two fractions?
2. How does the difference quotient relate to the derivative?
3. What happens to the difference quotient as \( h \to 0 \)?
4. How can you simplify a complex rational expression?
5. How do we interpret the result geometrically in terms of slopes?

### Tip:
When simplifying difference quotients, factor terms in the numerator before dividing by \( h \).

Simplify the expression f(x + h) and the difference quotient f(x + h) - f(x) / h for the function f(x) = 1 / (3x + 7).

This guide simplifies the expression f(x + h) and calculates the difference quotient for the function f(x) = 1 / (3x + 7). It covers key concepts such as algebra, the difference quotient, and functions, making it suitable for students in Grades 10-12.

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