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We are tasked with finding \( \frac{dy}{dx} \) using implicit differentiation for the equation:

\[
\sin(x) + \cos(y) = 3x - 2y
\]

### Step-by-step solution:

1. **Differentiate both sides with respect to \( x \):**
   - For the left-hand side \( \sin(x) + \cos(y) \):
     - \( \frac{d}{dx}[\sin(x)] = \cos(x) \)
     - \( \frac{d}{dx}[\cos(y)] = -\sin(y) \cdot \frac{dy}{dx} \) (using the chain rule)
     
   So, the derivative of the left-hand side is:
   \[
   \cos(x) - \sin(y) \cdot \frac{dy}{dx}
   \]

2. **Differentiate the right-hand side \( 3x - 2y \):**
   - \( \frac{d}{dx}[3x] = 3 \)
   - \( \frac{d}{dx}[-2y] = -2 \cdot \frac{dy}{dx} \) (again applying the chain rule)
   
   So, the derivative of the right-hand side is:
   \[
   3 - 2 \cdot \frac{dy}{dx}
   \]

3. **Combine the results:**
   \[
   \cos(x) - \sin(y) \cdot \frac{dy}{dx} = 3 - 2 \cdot \frac{dy}{dx}
   \]

4. **Solve for \( \frac{dy}{dx} \):**
   Rearranging the terms, move all terms involving \( \frac{dy}{dx} \) to one side:
   \[
   -\sin(y) \cdot \frac{dy}{dx} + 2 \cdot \frac{dy}{dx} = 3 - \cos(x)
   \]
   Factor out \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx}(-\sin(y) + 2) = 3 - \cos(x)
   \]
   Finally, solve for \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} = \frac{3 - \cos(x)}{2 - \sin(y)}
   \]

Thus, the solution is:

\[
\boxed{\frac{dy}{dx} = \frac{3 - \cos(x)}{2 - \sin(y)}}
\]

---

Do you want further details or have any questions?

Here are 5 related questions to consider:
1. How does implicit differentiation differ from explicit differentiation?
2. Can you apply this method to find higher-order derivatives?
3. How would the result change if the equation had more complex trigonometric functions?
4. How can we interpret the derivative geometrically in this context?
5. What are some common mistakes to avoid in implicit differentiation?

**Tip:** Always apply the chain rule when differentiating implicitly, especially when dealing with functions of \( y \) that depend on \( x \).

Find dy/dx by implicit differentiation. sin(x) + cos(y) = 3x - 2y

Learn how to solve the implicit differentiation problem: sin(x) + cos(y) = 3x - 2y. This problem involves applying the chain rule and differentiating trigonometric functions. Ideal for high school or college calculus students.

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