Find `dy/dx if 2y=e^y-2x`

Answer

To find \(\frac{dy}{dx}\) for the equation \(2y = e^y - 2x\), we will use implicit differentiation. Here are the steps:

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

Starting with the equation:
\[ 2y = e^y - 2x \]

We differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(2y) = \frac{d}{dx}(e^y) - \frac{d}{dx}(2x) \]

### Step 2: Apply the differentiation rules

Using the chain rule on the left side and the right side:
\[ 2 \frac{dy}{dx} = e^y \frac{dy}{dx} - 2 \]

### Step 3: Rearrange the equation

Now, we want to isolate \(\frac{dy}{dx}\). We can rearrange the equation:
\[ 2 \frac{dy}{dx} - e^y \frac{dy}{dx} = -2 \]

### Step 4: Factor out \(\frac{dy}{dx}\)

Factoring \(\frac{dy}{dx}\) from the left side gives us:
\[ \frac{dy}{dx} (2 - e^y) = -2 \]

### Step 5: Solve for \(\frac{dy}{dx}\)

Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{-2}{2 - e^y} \]

### Final Answer

Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = \frac{-2}{2 - e^y} \]

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