https://file.aiquickdraw.com/mathbot/file/1726752372375gtc8tp39.jpg

We are tasked with proving the equation:

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

### Step-by-step solution:

1. **Substitute \( f(x) = 5^x \) into the left-hand side:**

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

2. **Simplify the exponent \( 5^{x+h} \):**

   Using the property of exponents \( a^{m+n} = a^m \cdot a^n \), we can rewrite:

   \[
   5^{x+h} = 5^x \cdot 5^h
   \]

   So, the expression becomes:

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

3. **Factor out \( 5^x \):**

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

4. **Separate the terms:**

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

This matches the right-hand side of the given equation, proving the desired result.

### Final Answer:

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

---

Do you have any questions or would you like further details on this solution?

### Related questions:
1. How does this result relate to the definition of the derivative?
2. What would the derivative of \( f(x) = 5^x \) be using the limit definition?
3. How does this result change if \( f(x) = a^x \) for any constant \( a \)?
4. Can you find the derivative of \( f(x) = 5^x \) using logarithmic differentiation?
5. What happens when you take the second derivative of \( f(x) = 5^x \)?

### Tip:
Always remember that properties of exponents are key tools when dealing with exponential functions in calculus.

If f(x) = 5^x, show that (f(x + h) - f(x)) / h = 5^x ( (5^h - 1) / h )

Learn how to show that for the function f(x) = 5^x, the difference quotient (f(x + h) - f(x)) / h simplifies to 5^x ((5^h - 1) / h). This solution involves applying properties of exponents and algebraic manipulation, suitable for advanced high school or early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation