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We are tasked with finding \(\frac{dy}{dx}\) from the equation:

\[
\sin(xy) - \sin(2x) = 1
\]

Let’s differentiate both sides of the equation implicitly with respect to \(x\).

### Step 1: Differentiate each term
1. **Differentiate \(\sin(xy)\)**:
   - We need to apply the product rule because \(xy\) involves both \(x\) and \(y\).
   \[
   \frac{d}{dx} \left( \sin(xy) \right) = \cos(xy) \cdot \frac{d}{dx}(xy)
   \]
   Using the product rule on \(xy\), we get:
   \[
   \frac{d}{dx}(xy) = y + x \frac{dy}{dx}
   \]
   So, the derivative of \(\sin(xy)\) becomes:
   \[
   \cos(xy) \cdot (y + x \frac{dy}{dx})
   \]

2. **Differentiate \(\sin(2x)\)**:
   \[
   \frac{d}{dx} \left( \sin(2x) \right) = 2 \cos(2x)
   \]

3. **Differentiate the constant \(1\)**:
   \[
   \frac{d}{dx}(1) = 0
   \]

### Step 2: Write the full derivative equation
After differentiating both sides, we have:
\[
\cos(xy) \cdot (y + x \frac{dy}{dx}) - 2\cos(2x) = 0
\]

### Step 3: Solve for \(\frac{dy}{dx}\)
Rearrange the equation to isolate \(\frac{dy}{dx}\):
\[
\cos(xy) \cdot (y + x \frac{dy}{dx}) = 2 \cos(2x)
\]

Expand the left-hand side:
\[
\cos(xy) \cdot y + \cos(xy) \cdot x \frac{dy}{dx} = 2 \cos(2x)
\]

Now, isolate \(\frac{dy}{dx}\):
\[
\cos(xy) \cdot x \frac{dy}{dx} = 2 \cos(2x) - \cos(xy) \cdot y
\]
\[
\frac{dy}{dx} = \frac{2 \cos(2x) - \cos(xy) \cdot y}{x \cos(xy)}
\]

### Step 4: Match with the options
Now compare this expression with the given choices:

- **Option C** matches this result:
  \[
  \frac{2 \cos(2x) - y \cos(xy)}{x \cos(xy)}
  \]

Thus, the correct answer is **C**.

Would you like any further details or have any questions? Here are some related questions for deeper understanding:

1. How do you apply implicit differentiation in other contexts?
2. What is the product rule, and how is it applied here?
3. Why do we differentiate \(\sin(2x)\) using the chain rule?
4. Can you explain how to isolate \(\frac{dy}{dx}\) in a different equation?
5. What would happen if the equation had more complex trigonometric functions?

**Tip**: Always apply the product rule carefully when differentiating expressions like \(\sin(xy)\), which involve products of variables.

Tentukan \(\frac{dy}{dx}\) dari persamaan \(\sin(xy) - \sin(2x) = 1\)?

Learn how to find \(\frac{dy}{dx}\) from the equation \(\sin(xy) - \sin(2x) = 1\) using implicit differentiation. This problem involves applying the product rule and chain rule, making it suitable for college-level calculus students.

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