https://file.aiquickdraw.com/mathbot/file/1728273481130r4rzg8z3.jpg

To solve for \( f'(x) \), where \( f(x) = 2 \sin(x) + \ln(5x) \), we need to take the derivative of each term individually.

1. **Derivative of \( 2 \sin(x) \)**:
   \[
   \frac{d}{dx}[2 \sin(x)] = 2 \cos(x)
   \]

2. **Derivative of \( \ln(5x) \)**:
   Using the chain rule, the derivative of \( \ln(5x) \) is:
   \[
   \frac{d}{dx}[\ln(5x)] = \frac{1}{5x} \cdot 5 = \frac{1}{x}
   \]

So, combining these, the derivative \( f'(x) \) is:
\[
f'(x) = 2 \cos(x) + \frac{1}{x}
\]

Would you like further details on the graph or have any questions?

Here are some related questions to explore further:
1. What is the second derivative \( f''(x) \) of the function?
2. How would the graph of \( f(x) \) and \( f'(x) \) differ visually?
3. How does the term \( \ln(5x) \) affect the behavior of the derivative?
4. What are the critical points of \( f'(x) \)?
5. How do we find the inflection points of \( f(x) \)?

**Tip**: When applying the chain rule, always check if constants inside the logarithm or trigonometric functions need to be factored out!

Learn how to differentiate the function f(x) = 2 sin(x) + ln(5x). This problem involves trigonometric and logarithmic differentiation, applying the chain rule, and finding the derivative step by step. Suitable for high school calculus students (Grades 11-12).

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