The problem asks to differentiate the function:

$f(x) = x^6 + 5^x$

Step-by-step differentiation:

1. Differentiate $x^6$: $\frac{d}{dx}(x^6) = 6x^5$

2. Differentiate $5^x$: To differentiate an exponential function with a base other than $e$, use the rule: $\frac{d}{dx}(a^x) = a^x \ln(a)$ Here, $a = 5$, so: $\frac{d}{dx}(5^x) = 5^x \ln(5)$

Combine the results:

$f'(x) = 6x^5 + 5^x \ln(5)$

So, the derivative is:

$f'(x) = 6x^5 + 5^x \ln(5)$

Would you like more details on any of these steps or have any questions? Here are a few related problems to expand your understanding:

1. What is the derivative of $7^x$?
2. Can you differentiate $f(x) = x^7 + 3^x$?
3. How do you differentiate $x^n$ for any integer $n$?
4. What is the derivative of $e^x$?
5. Can you apply the product rule to $x^3 \cdot 2^x$?

Tip: When differentiating exponential functions, always remember to multiply by the natural logarithm of the base.