Find the derivative of the following w.r.t. x :
`2x+3y=siny`

Answer

To find the derivative of the equation \( 2x + 3y = \sin y \) with respect to \( x \), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides with respect to \( x \)

We start with the equation:

\[ 2x + 3y = \sin y \]

Now, we differentiate both sides with respect to \( x \):

\[ \frac{d}{dx}(2x) + \frac{d}{dx}(3y) = \frac{d}{dx}(\sin y) \]

### Step 2: Apply differentiation rules

Using the differentiation rules, we have:

- The derivative of \( 2x \) is \( 2 \).
- For \( 3y \), we use the chain rule: \( \frac{d}{dx}(3y) = 3 \frac{dy}{dx} \).
- For \( \sin y \), we also use the chain rule: \( \frac{d}{dx}(\sin y) = \cos y \frac{dy}{dx} \).

Putting this all together, we get:

\[ 2 + 3 \frac{dy}{dx} = \cos y \frac{dy}{dx} \]

### Step 3: Rearrange the equation

Now, we will rearrange the equation to isolate \( \frac{dy}{dx} \):

\[ 3 \frac{dy}{dx} - \cos y \frac{dy}{dx} = -2 \]

Factoring out \( \frac{dy}{dx} \):

\[ \frac{dy}{dx}(3 - \cos y) = -2 \]

### 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 of \( y \) with respect to \( x \) is:

\[ \frac{dy}{dx} = \frac{-2}{3 - \cos y} \]