Find `dy/dx if xcosx=2siny`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x \cos x = 2 \sin y\), we will use implicit differentiation. Here are the steps to solve the problem:

### Step 1: Differentiate both sides with respect to \(x\)
We start with the equation:
\[ x \cos x = 2 \sin y \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the Product Rule on the left side
Using the product rule, where \(u = x\) and \(v = \cos x\):
\[ \frac{d}{dx}(x \cos x) = \frac{dx}{dx} \cdot \cos x + x \cdot \frac{d}{dx}(\cos x) \]
This gives us:
\[ \cos x + x(-\sin x) = \cos x - x \sin x \]

### Step 3: Differentiate the right side
For the right side, we differentiate \(2 \sin y\):
\[ \frac{d}{dx}(2 \sin y) = 2 \cos y \cdot \frac{dy}{dx} \]

### Step 4: Set the derivatives equal to each other
Now we equate the derivatives from both sides:
\[ \cos x - x \sin x = 2 \cos y \cdot \frac{dy}{dx} \]

### Step 5: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we rearrange the equation:
\[ \frac{dy}{dx} = \frac{\cos x - x \sin x}{2 \cos y} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{\cos x - x \sin x}{2 \cos y} \]