Find `dy/dx if x-9y=tany`

Answer

To find \(\frac{dy}{dx}\) for the equation \(x - 9y = \tan(y)\), we will differentiate both sides with respect to \(x\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ x - 9y = \tan(y) \]

2. **Differentiate both sides with respect to \(x\):**
- The left side:
- The derivative of \(x\) with respect to \(x\) is \(1\).
- The derivative of \(-9y\) with respect to \(x\) is \(-9 \frac{dy}{dx}\) (using the chain rule).
- The right side:
- The derivative of \(\tan(y)\) with respect to \(x\) is \(\sec^2(y) \frac{dy}{dx}\) (using the chain rule).

Putting it all together, we have:
\[ 1 - 9 \frac{dy}{dx} = \sec^2(y) \frac{dy}{dx} \]

3. **Rearrange the equation to isolate \(\frac{dy}{dx}\):**
\[ 1 = \sec^2(y) \frac{dy}{dx} + 9 \frac{dy}{dx} \]
\[ 1 = \left(\sec^2(y) + 9\right) \frac{dy}{dx} \]

4. **Solve for \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{1}{\sec^2(y) + 9} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{1}{\sec^2(y) + 9} \]