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

Answer

To find \( \frac{dy}{dx} \) for the equation \( 3x - 2y = \tan x \), we will differentiate both sides of the equation with respect to \( x \). Here are the steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 3x - 2y = \tan x \]

2. **Differentiate both sides with respect to \( x \):**
- The derivative of \( 3x \) is \( 3 \).
- The derivative of \( -2y \) is \( -2 \frac{dy}{dx} \) (using the chain rule).
- The derivative of \( \tan x \) is \( \sec^2 x \).

Thus, differentiating gives us:
\[ \frac{d}{dx}(3x) - \frac{d}{dx}(2y) = \frac{d}{dx}(\tan x) \]
Which simplifies to:
\[ 3 - 2 \frac{dy}{dx} = \sec^2 x \]

3. **Rearrange the equation to solve for \( \frac{dy}{dx} \):**
\[ -2 \frac{dy}{dx} = \sec^2 x - 3 \]
Now, divide both sides by -2:
\[ \frac{dy}{dx} = \frac{3 - \sec^2 x}{2} \]

### Final Result:
\[ \frac{dy}{dx} = \frac{3 - \sec^2 x}{2} \]