If `sin(x y)+cos(x y)=0` , then `(dy)/(dx)` is

Options

A. `(y)/(x)`
B. `-(y)/(x)`
C. `-(x)/(y)`
D. `(x)/(y)`

Answer

To find \(\frac{dy}{dx}\) given the equation \( \sin(xy) + \cos(xy) = 0 \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin(xy) + \cos(xy) = 0 \]
We can rearrange this to isolate \(\sin(xy)\):
\[ \sin(xy) = -\cos(xy) \]

### Step 2: Divide by \(\cos(xy)\)
Next, we can divide both sides by \(\cos(xy)\) (assuming \(\cos(xy) \neq 0\)):
\[ \frac{\sin(xy)}{\cos(xy)} = -1 \]
This simplifies to:
\[ \tan(xy) = -1 \]

### Step 3: Solve for \(xy\)
The equation \(\tan(xy) = -1\) implies that:
\[ xy = \frac{3\pi}{4} + n\pi \quad \text{for } n \in \mathbb{Z} \]

### Step 4: Differentiate both sides
Now, we differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(xy) = \frac{d}{dx}\left(\frac{3\pi}{4} + n\pi\right) \]
Since the right-hand side is constant with respect to \(x\), its derivative is 0:
\[ \frac{d}{dx}(xy) = 0 \]

### Step 5: Apply the product rule
Using the product rule on the left-hand side:
\[ x \frac{dy}{dx} + y = 0 \]

### Step 6: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ x \frac{dy}{dx} = -y \]
\[ \frac{dy}{dx} = -\frac{y}{x} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = -\frac{y}{x} \]

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