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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.

Differentiate the following function: f(x) = x^6 + 5^x

Learn how to differentiate the function f(x) = x^6 + 5^x using the power rule and the derivative of exponential functions. This solution includes step-by-step differentiation of each term and combines the results for a complete derivative. Suitable for undergraduate calculus students.

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