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To estimate \( f'(2) \), we can use the definition of the derivative as a limit:

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

For this problem, we'll estimate \( f'(2) \) using a small value for \( h \), which is often something like 0.01 or 0.001. Let's use \( h = 0.001 \) to get a good approximation.

Given \( f(x) = 5^x \), we need to calculate the following:

\[
f'(2) \approx \frac{f(2 + 0.001) - f(2)}{0.001}
\]

First, calculate \( f(2) \) and \( f(2 + 0.001) \):

\[
f(2) = 5^2 = 25
\]
\[
f(2 + 0.001) = 5^{2.001}
\]

I'll now calculate \( 5^{2.001} \) and then approximate the derivative.Using \( h = 0.001 \), we estimate that:

\[
f'(2) \approx 40.27
\]

This is the approximation for the derivative of \( f(x) = 5^x \) at \( x = 2 \), rounded to two decimal places.

Would you like more details or further explanation? Here are five related questions to explore:

1. How does the value of \( h \) affect the accuracy of the derivative estimate?
2. What is the exact derivative of \( f(x) = 5^x \)?
3. How can we estimate derivatives for more complicated functions?
4. Can we estimate higher-order derivatives using similar methods?
5. What are some common applications of derivatives in real-world problems?

**Tip:** When estimating derivatives numerically, using a smaller \( h \) generally improves accuracy but may lead to rounding errors in calculations.

Let f(x) = 5^x. Use a small interval to estimate f'(2). Round your answer to two decimal places.

Estimate the derivative of the exponential function f(x) = 5^x at x = 2 using a small interval approximation. Follow this step-by-step solution, using f'(x) ≈ (f(x + h) - f(x)) / h, and round the result to two decimal places. Suitable for Grades 10-12.

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