Find `dy/dx if 2y-e^x=6`

Answer

To find \( \frac{dy}{dx} \) for the equation \( 2y - e^x = 6 \), we will follow these steps:

### Step 1: Start with the given equation
The given equation is:
\[ 2y - e^x = 6 \]

### Step 2: Differentiate both sides with respect to \( x \)
We will differentiate the left-hand side and the right-hand side with respect to \( x \):
\[ \frac{d}{dx}(2y) - \frac{d}{dx}(e^x) = \frac{d}{dx}(6) \]

### Step 3: Apply differentiation rules
Using the chain rule for \( 2y \) and the fact that the derivative of \( e^x \) is \( e^x \), we get:
\[ 2 \frac{dy}{dx} - e^x = 0 \]

### Step 4: Solve for \( \frac{dy}{dx} \)
Now, we can isolate \( \frac{dy}{dx} \):
\[ 2 \frac{dy}{dx} = e^x \]
\[ \frac{dy}{dx} = \frac{e^x}{2} \]

### Final Answer
Thus, the value of \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{e^x}{2} \]

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