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We are asked to find the derivative of the function:

\[
f(x) = \ln(5) + 0.6^x + \ln(x)
\]

Let’s break it down and apply the rules of differentiation:

1. \(\ln(5)\) is a constant, and the derivative of a constant is zero:
   \[
   \frac{d}{dx}[\ln(5)] = 0
   \]

2. \(0.6^x\) can be treated as an exponential function of the form \(a^x\). The derivative of \(a^x\) with respect to \(x\) is:
   \[
   \frac{d}{dx}[a^x] = a^x \ln(a)
   \]
   So,
   \[
   \frac{d}{dx}[0.6^x] = 0.6^x \ln(0.6)
   \]

3. \(\ln(x)\) is a standard logarithmic function, and the derivative of \(\ln(x)\) is:
   \[
   \frac{d}{dx}[\ln(x)] = \frac{1}{x}
   \]

Now, combining all the derivatives, we get the total derivative of \(f(x)\):
\[
f'(x) = 0 + 0.6^x \ln(0.6) + \frac{1}{x}
\]

Thus, the derivative of \(f(x)\) is:
\[
f'(x) = 0.6^x \ln(0.6) + \frac{1}{x}
\]

Would you like further details or explanations on any part of this? 

Here are five related questions to deepen your understanding:
1. How do we differentiate exponential functions in general?
2. What is the importance of the natural logarithm in calculus?
3. Can we generalize the derivative of \(a^x\) for any base \(a\)?
4. How would the derivative change if we had a function like \(f(x) = 0.6x + \ln(x)\)?
5. What is the difference between differentiating constant logarithms and variable logarithms?

**Tip:** Always remember that the derivative of a constant is zero, regardless of its complexity.

Find the derivative of the function f(x) = ln(5) + 0.6^x + ln(x).

Learn how to find the derivative of f(x) = ln(5) + 0.6^x + ln(x). This problem covers key calculus concepts, including the derivative of logarithmic and exponential functions. Suitable for high school students in Grades 11-12.

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