https://r2.erweima.ai/mathbot/image/1726795490897y22uqij4.jpg

We are tasked with finding the derivative of the function given in the image:

\[
y = 3 \ln(x) + \sin\left(\frac{\pi}{7}\right)
\]

### Step-by-Step Derivative Process

1. **Identify the terms**:
   - The function is composed of two parts: \( 3 \ln(x) \) and \( \sin\left(\frac{\pi}{7}\right) \).
   - The term \( 3 \ln(x) \) depends on \( x \), while \( \sin\left(\frac{\pi}{7}\right) \) is a constant since \( \frac{\pi}{7} \) is just a number.

2. **Differentiate each term**:
   - **Derivative of \( 3 \ln(x) \)**:
     \[
     \frac{d}{dx}[3 \ln(x)] = 3 \cdot \frac{1}{x} = \frac{3}{x}
     \]
   - **Derivative of \( \sin\left(\frac{\pi}{7}\right) \)**:
     Since this is a constant (because \( \frac{\pi}{7} \) is a constant), the derivative of any constant is 0:
     \[
     \frac{d}{dx} \left[ \sin\left(\frac{\pi}{7}\right) \right] = 0
     \]

3. **Combine the results**:
   \[
   \frac{dy}{dx} = \frac{3}{x} + 0 = \frac{3}{x}
   \]

Thus, the derivative of the given function is:

\[
\boxed{\frac{dy}{dx} = \frac{3}{x}}
\]

Would you like more details or clarifications? Here are some follow-up questions you might consider:

1. How do we differentiate logarithmic functions?
2. Why is \( \sin\left(\frac{\pi}{7}\right) \) treated as a constant in this case?
3. What happens when differentiating other trigonometric functions?
4. Can we apply the chain rule to more complex logarithmic functions?
5. How do you integrate the result of \( \frac{3}{x} \)?

**Tip**: Always isolate constants when taking derivatives to simplify the process.

Find the derivative of y = 3ln(x) + sin(π/7) with a clear step-by-step guide. Learn how to differentiate logarithmic and constant trigonometric functions. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation