Given the function \( f(x) = 5^x \), we want to find the expression for the difference quotient:

\[
\frac{f(x+h) - f(x)}{h}
\]

### Step-by-Step Calculation:

1. **Substitute the Function:**

   Substitute \( f(x) = 5^x \) into the difference quotient:

   \[
   \frac{5^{x+h} - 5^x}{h}
   \]

2. **Apply Properties of Exponents:**

   Recall that \( 5^{x+h} = 5^x \cdot 5^h \). Using this, we rewrite the expression:

   \[
   \frac{5^x \cdot 5^h - 5^x}{h}
   \]

3. **Factor Out Common Terms:**

   Factor out \( 5^x \) from the numerator:

   \[
   = \frac{5^x (5^h - 1)}{h}
   \]

### Simplified Difference Quotient:

The difference quotient for \( f(x) = 5^x \) is:

\[
\frac{f(x+h) - f(x)}{h} = \frac{5^x (5^h - 1)}{h}
\]

### Interpretation:

- This expression represents the average rate of change of \( 5^x \) over an interval of \( h \).
- If you take the limit as \( h \to 0 \), this expression will give the derivative of \( f(x) = 5^x \).

Would you like to see the derivative calculation using this quotient?

### Related Questions:
1. How do you find the derivative of \( 5^x \) using limits?
2. Why is the difference quotient important in calculus?
3. How does the expression change if \( f(x) = a^x \) for a different base \( a \)?
4. What is the derivative of exponential functions in general?
5. What happens to the difference quotient as \( h \to 0 \)?

**Tip:** Knowing the difference quotient is fundamental in understanding how a function changes, leading to the concept of the derivative in calculus.

Explore the difference quotient for the function f(x) = 5^x. Learn how to calculate (f(x+h) - f(x)) / h step-by-step with examples. This concept is fundamental in calculus and helps determine the derivative of exponential functions. Suitable for advanced high school students.

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