Statement 1: If `e^(xy)+ln(xy)+cos(xy)+5=0,` then `(dy)/(dx)=-y/x`. Statement 2: `d/(dx)(xy)=0,y` is a function of `x implies(dy)/(dx)=-y/x`.

Options

A. Statement 1: is True, Statement 2 is True , Statement 2 is a correct explanation for statement 1
B. Statement 1 is True, Statement 2 is True Statement 2 is NOT a correct explanation for Statement 1
C. Statement 1 is True, Statement 2 is False
D. Statement 1 is False, Statement 2 is True

Explanation

To solve the problem, we need to analyze the statements provided and differentiate the given equation. Let's break it down step by step.

### Step 1: Differentiate the given equation
We start with the equation:
\[ e^{xy} + \ln(xy) + \cos(xy) + 5 = 0 \]

We will differentiate both sides with respect to \( x \).

### Step 2: Apply the chain rule and product rule
Differentiating each term:
1. For \( e^{xy} \):
\[ \frac{d}{dx}(e^{xy}) = e^{xy} \cdot \frac{d}{dx}(xy) = e^{xy} \cdot (y + x \frac{dy}{dx}) \]
2. For \( \ln(xy) \):
\[ \frac{d}{dx}(\ln(xy)) = \frac{1}{xy} \cdot \frac{d}{dx}(xy) = \frac{1}{xy} \cdot (y + x \frac{dy}{dx}) \]
3. For \( \cos(xy) \):
\[ \frac{d}{dx}(\cos(xy)) = -\sin(xy) \cdot \frac{d}{dx}(xy) = -\sin(xy) \cdot (y + x \frac{dy}{dx}) \]
4. The derivative of the constant \( 5 \) is \( 0 \).

Putting it all together, we have:
\[ e^{xy}(y + x \frac{dy}{dx}) + \frac{1}{xy}(y + x \frac{dy}{dx}) - \sin(xy)(y + x \frac{dy}{dx}) = 0 \]

### Step 3: Factor out common terms
We can factor out \( (y + x \frac{dy}{dx}) \):
\[ (y + x \frac{dy}{dx}) \left( e^{xy} + \frac{1}{xy} - \sin(xy) \right) = 0 \]

### Step 4: Analyze the factors
This gives us two possibilities:
1. \( y + x \frac{dy}{dx} = 0 \)
2. \( e^{xy} + \frac{1}{xy} - \sin(xy) = 0 \)

Since we are interested in the first case:
\[ y + x \frac{dy}{dx} = 0 \]
This can be rearranged to find \( \frac{dy}{dx} \):
\[ x \frac{dy}{dx} = -y \implies \frac{dy}{dx} = -\frac{y}{x} \]

### Conclusion for Statement 1
Thus, Statement 1 is true:
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Step 5: Analyze Statement 2
Statement 2 states:
\[ \frac{d}{dx}(xy) = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
If \( \frac{d}{dx}(xy) = 0 \), it implies that \( xy \) is constant. Differentiating \( xy \) gives:
\[ y + x \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{y}{x} \]
This is consistent with our earlier result.

### Final Conclusion
Both statements are true, and Statement 2 correctly explains Statement 1. Therefore, the correct option is:
**Option 1: Statement 1 is true, Statement 2 is true, and Statement 2 is a correct explanation of Statement 1.**