Given the function $f(x) = e^x + (5)^x$, we need to find the value of its derivative at $x=0$, i.e., $f'(0)$.

1. Problem Description

Given the function $f(x) = e^x + (5)^x$, we need to find the value of its derivative at $x=0$, i.e., $f'(0)$.

2. Solution Steps

First, we need to find the derivative of the function $f(x)$.

The derivative of $e^x$ is $e^x$.

The derivative of $a^x$ is $a^x \ln(a)$.

Therefore, the derivative of $5^x$ is $5^x \ln(5)$.

So, the derivative of $f(x)$ is:

$f'(x) = \frac{d}{dx}(e^x + 5^x) = \frac{d}{dx}(e^x) + \frac{d}{dx}(5^x) = e^x + 5^x \ln(5)$

Now, we need to evaluate $f'(0)$:

$f'(0) = e^0 + 5^0 \ln(5)$

Since $e^0 = 1$ and $5^0 = 1$, we have:

$f'(0) = 1 + 1 \cdot \ln(5) = 1 + \ln(5)$

3. Final Answer

$f'(0) = 1 + \ln(5)$