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

Answer

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

### Step-by-Step Solution:

1. **Differentiate both sides:**
Start with the equation:
\[ 2x - 3y = \log y \]
Differentiate both sides with respect to \(x\):
\[ \frac{d}{dx}(2x) - \frac{d}{dx}(3y) = \frac{d}{dx}(\log y) \]

2. **Apply the differentiation rules:**
- The derivative of \(2x\) is \(2\).
- The derivative of \(3y\) using the chain rule is \(3\frac{dy}{dx}\).
- The derivative of \(\log y\) using the chain rule is \(\frac{1}{y}\frac{dy}{dx}\).

Thus, we have:
\[ 2 - 3\frac{dy}{dx} = \frac{1}{y}\frac{dy}{dx} \]

3. **Rearrange the equation:**
Move all terms involving \(\frac{dy}{dx}\) to one side:
\[ 2 = 3\frac{dy}{dx} + \frac{1}{y}\frac{dy}{dx} \]

4. **Factor out \(\frac{dy}{dx}\):**
\[ 2 = \left(3 + \frac{1}{y}\right)\frac{dy}{dx} \]

5. **Solve for \(\frac{dy}{dx}\):**
Divide both sides by \(\left(3 + \frac{1}{y}\right)\):
\[ \frac{dy}{dx} = \frac{2}{3 + \frac{1}{y}} \]

6. **Simplify the expression:**
To simplify, multiply the numerator and denominator by \(y\):
\[ \frac{dy}{dx} = \frac{2y}{3y + 1} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{2y}{3y + 1} \]