If ` 3 sin (xy) + 4 cos (xy) = 5 , " then " (dy)/(dx) ` is equal to

Options

A. `(-y)/(x)`
B. `(3 sin (xy) + 4 cos (xy))/(3 cos (xy) - 4 sin (xy))`
C. `(3 ocs (xy) + 4 sin (xy))/(4 cos (xy) - 3 sin (xy))`
D. None of these

Explanation

To solve the problem, we need to differentiate the equation \( 3 \sin(xy) + 4 \cos(xy) = 5 \) with respect to \( x \) and find \( \frac{dy}{dx} \).

### Step-by-Step Solution:

1. **Differentiate both sides with respect to \( x \)**:
\[ \frac{d}{dx}(3 \sin(xy)) + \frac{d}{dx}(4 \cos(xy)) = \frac{d}{dx}(5) \]

2. **Apply the chain rule**:
- For \( 3 \sin(xy) \):
\[ 3 \cos(xy) \cdot \frac{d}{dx}(xy) = 3 \cos(xy) \cdot (y + x \frac{dy}{dx}) \]
- For \( 4 \cos(xy) \):
\[ 4 (-\sin(xy)) \cdot \frac{d}{dx}(xy) = -4 \sin(xy) \cdot (y + x \frac{dy}{dx}) \]
- The derivative of a constant (5) is 0.

3. **Combine the derivatives**:
\[ 3 \cos(xy) (y + x \frac{dy}{dx}) - 4 \sin(xy) (y + x \frac{dy}{dx}) = 0 \]

4. **Distribute and rearrange**:
\[ (3 \cos(xy) - 4 \sin(xy))(y + x \frac{dy}{dx}) = 0 \]

5. **Set the factor equal to zero** (since \( y + x \frac{dy}{dx} \) cannot be zero):
\[ 3 \cos(xy) - 4 \sin(xy) = 0 \]

6. **Isolate \( \frac{dy}{dx} \)**:
- Rearranging gives:
\[ 3 \cos(xy) y + 3x \cos(xy) \frac{dy}{dx} - 4 \sin(xy) y - 4x \sin(xy) \frac{dy}{dx} = 0 \]
- Grouping \( \frac{dy}{dx} \):
\[ (3x \cos(xy) - 4x \sin(xy)) \frac{dy}{dx} = 4y \sin(xy) - 3y \cos(xy) \]

7. **Solve for \( \frac{dy}{dx} \)**:
\[ \frac{dy}{dx} = \frac{4y \sin(xy) - 3y \cos(xy)}{3x \cos(xy) - 4x \sin(xy)} \]

8. **Factor out common terms**:
\[ \frac{dy}{dx} = \frac{y(4 \sin(xy) - 3 \cos(xy))}{x(3 \cos(xy) - 4 \sin(xy))} \]

9. **Final simplification**:
- If we observe the numerator and denominator, we can simplify:
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Conclusion:
Thus, the final answer is:
\[ \frac{dy}{dx} = -\frac{y}{x} \]