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Sure, let's find the fourth derivative of the function \( f(x) = 5^x + \ln(x) - 195x \).

We'll go through the process step by step.

1. **First Derivative:**
   \[
   f(x) = 5^x + \ln(x) - 195x
   \]
   The derivative of \( 5^x \) is \( 5^x \ln(5) \). The derivative of \( \ln(x) \) is \( \frac{1}{x} \). The derivative of \( -195x \) is \( -195 \).

   So, the first derivative \( f'(x) \) is:
   \[
   f'(x) = 5^x \ln(5) + \frac{1}{x} - 195
   \]

2. **Second Derivative:**
   \[
   f'(x) = 5^x \ln(5) + \frac{1}{x} - 195
   \]
   The derivative of \( 5^x \ln(5) \) is \( 5^x (\ln(5))^2 \). The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).

   So, the second derivative \( f''(x) \) is:
   \[
   f''(x) = 5^x (\ln(5))^2 - \frac{1}{x^2}
   \]

3. **Third Derivative:**
   \[
   f''(x) = 5^x (\ln(5))^2 - \frac{1}{x^2}
   \]
   The derivative of \( 5^x (\ln(5))^2 \) is \( 5^x (\ln(5))^3 \). The derivative of \( -\frac{1}{x^2} \) is \( \frac{2}{x^3} \).

   So, the third derivative \( f'''(x) \) is:
   \[
   f'''(x) = 5^x (\ln(5))^3 + \frac{2}{x^3}
   \]

4. **Fourth Derivative:**
   \[
   f'''(x) = 5^x (\ln(5))^3 + \frac{2}{x^3}
   \]
   The derivative of \( 5^x (\ln(5))^3 \) is \( 5^x (\ln(5))^4 \). The derivative of \( \frac{2}{x^3} \) is \( -\frac{6}{x^4} \).

   So, the fourth derivative \( f^{(4)}(x) \) is:
   \[
   f^{(4)}(x) = 5^x (\ln(5))^4 - \frac{6}{x^4}
   \]

Thus, the fourth derivative of \( f(x) = 5^x + \ln(x) - 195x \) is:
\[
f^{(4)}(x) = 5^x (\ln(5))^4 - \frac{6}{x^4}
\]

Do you want more details on any specific step or have any questions?

Here are 8 relative questions to consider:
1. What is the general formula for the derivative of an exponential function?
2. How does the chain rule apply to the differentiation of \(5^x\)?
3. What is the process for finding the nth derivative of a function?
4. Can you explain the significance of logarithmic differentiation?
5. How do you differentiate logarithmic functions like \(\ln(x)\)?
6. What are some real-world applications of higher-order derivatives?
7. How can higher-order derivatives be used in Taylor series expansions?
8. What techniques can be used to simplify the differentiation of complex functions?

**Tip:** When dealing with higher-order derivatives, it is often helpful to look for patterns in the derivatives to simplify the process.

Learn how to find the fourth derivative of the function f(x) = 5^x + ln(x) - 195x step-by-step. This problem involves differentiation of exponential and logarithmic functions, including the use of higher-order derivatives. Suitable for advanced mathematics students.

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