Find `dy/dx if 2x+3y= siny`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x + 3y = \sin y\), 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:
\[ 2x + 3y = \sin y \]
Now, we differentiate each term with respect to \(x\).

### Step 2: Apply the differentiation rules
- The derivative of \(2x\) with respect to \(x\) is \(2\).
- The derivative of \(3y\) with respect to \(x\) is \(3 \frac{dy}{dx}\) (using the chain rule).
- The derivative of \(\sin y\) with respect to \(x\) is \(\cos y \frac{dy}{dx}\) (using the chain rule).

Putting this together, we have:
\[ \frac{d}{dx}(2x) + \frac{d}{dx}(3y) = \frac{d}{dx}(\sin y) \]
This gives:
\[ 2 + 3 \frac{dy}{dx} = \cos y \frac{dy}{dx} \]

### Step 3: Rearrange the equation
Now, we want to isolate \(\frac{dy}{dx}\). We can rearrange the equation:
\[ 3 \frac{dy}{dx} - \cos y \frac{dy}{dx} = -2 \]
Factoring out \(\frac{dy}{dx}\) from the left side:
\[ \frac{dy}{dx}(3 - \cos y) = -2 \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{-2}{3 - \cos y} \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{-2}{3 - \cos y} \]

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