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

Answer

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

### Step-by-Step Solution:

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

2. **Differentiate both sides with respect to \(x\):**
- The left side is simply \(\frac{dx}{dx} = 1\).
- For the right side, we apply the product rule and the chain rule:
- The derivative of \(3y\) with respect to \(x\) is \(3 \frac{dy}{dx}\) (since \(y\) is a function of \(x\)).
- The derivative of \(-\tan x\) with respect to \(x\) is \(-\sec^2 x\).

Thus, differentiating gives:
\[ 1 = 3 \frac{dy}{dx} - \sec^2 x \]

3. **Rearrange the equation to solve for \(\frac{dy}{dx}\):**
\[ 3 \frac{dy}{dx} = 1 + \sec^2 x \]

4. **Divide both sides by 3 to isolate \(\frac{dy}{dx}\):**
\[ \frac{dy}{dx} = \frac{1 + \sec^2 x}{3} \]

### Final Result:
\[ \frac{dy}{dx} = \frac{1 + \sec^2 x}{3} \]