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.