https://file.aiquickdraw.com/mathbot/file/1725460633449y9htf974.jpg

The image you uploaded shows a math problem. Here's the content of the question:

\[ y = \ln(x) + 6x, \quad \text{Find} \ \frac{dy}{dx} + y. \]

The problem asks for the derivative of \( y \) with respect to \( x \) and then adds the original function \( y \) to this derivative.

### Step-by-Step Solution:

1. **Find the derivative of \( y \) with respect to \( x \):**
   \[
   \frac{dy}{dx} = \frac{d}{dx} \left(\ln(x) + 6x\right)
   \]
   The derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \).
   The derivative of \( 6x \) with respect to \( x \) is \( 6 \).
   \[
   \frac{dy}{dx} = \frac{1}{x} + 6
   \]

2. **Add \( y \) to \( \frac{dy}{dx} \):**
   \[
   \frac{dy}{dx} + y = \left(\frac{1}{x} + 6\right) + \left(\ln(x) + 6x\right)
   \]
   Simplify the expression:
   \[
   \frac{dy}{dx} + y = \frac{1}{x} + 6 + \ln(x) + 6x
   \]

3. **Final Answer:**
   \[
   \frac{dy}{dx} + y = \frac{1}{x} + \ln(x) + 6x + 6
   \]

So, the correct answer corresponds to the option that represents this expression.

Would you like more details on any part of the solution, or do you have any questions?

### Related Questions:
1. How do you differentiate logarithmic functions in general?
2. What are the basic rules of differentiation?
3. How do you find the derivative of a sum of functions?
4. What are some practical applications of derivatives in real life?
5. How does the derivative of a product of functions differ from the sum of functions?

**Tip:** When finding the derivative of a logarithmic function, always remember that the derivative of \( \ln(x) \) with respect to \( x \) is \( \frac{1}{x} \).

Understand how to find the derivative of the function y = ln(x) + 6x and how to add it to the original function. This problem covers the differentiation of logarithmic functions and applies the sum rule. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation