Find `dy/dx if 2x-3sinx=2y`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2x - 3\sin(x) = 2y\), we will use implicit differentiation. Here’s a step-by-step solution:

### Step 1: Differentiate both sides of the equation
We start with the equation:
\[ 2x - 3\sin(x) = 2y \]
Now, we differentiate both sides with respect to \(x\).

### Step 2: Apply the differentiation
Differentiating the left side:
- The derivative of \(2x\) is \(2\).
- The derivative of \(-3\sin(x)\) is \(-3\cos(x)\) (using the chain rule).

So, the left side becomes:
\[ \frac{d}{dx}(2x - 3\sin(x)) = 2 - 3\cos(x) \]

Now, differentiate the right side:
- The derivative of \(2y\) with respect to \(x\) is \(2\frac{dy}{dx}\) (using the chain rule).

So, the right side becomes:
\[ \frac{d}{dx}(2y) = 2\frac{dy}{dx} \]

### Step 3: Set the derivatives equal to each other
Now we equate the derivatives from both sides:
\[ 2 - 3\cos(x) = 2\frac{dy}{dx} \]

### Step 4: Solve for \(\frac{dy}{dx}\)
To isolate \(\frac{dy}{dx}\), we divide both sides by \(2\):
\[ \frac{dy}{dx} = \frac{2 - 3\cos(x)}{2} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = 1 - \frac{3}{2}\cos(x) \]

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