Find `dy/dx` if `sec(x+y) = xy`

Explanation

To find \( \frac{dy}{dx} \) for the equation \( \sec(x+y) = xy \), we will differentiate both sides of the equation with respect to \( x \) and then solve for \( \frac{dy}{dx} \).

### Step-by-Step Solution:

1. **Differentiate Both Sides:**
\[ \frac{d}{dx}[\sec(x+y)] = \frac{d}{dx}[xy] \]

2. **Apply the Chain Rule on the Left Side:**
The derivative of \( \sec(u) \) is \( \sec(u) \tan(u) \cdot \frac{du}{dx} \). Here, \( u = x + y \).
\[ \frac{d}{dx}[\sec(x+y)] = \sec(x+y) \tan(x+y) \cdot \frac{d}{dx}(x+y) \]
Now differentiate \( x+y \):
\[ \frac{d}{dx}(x+y) = 1 + \frac{dy}{dx} \]
So, we have:
\[ \sec(x+y) \tan(x+y) (1 + \frac{dy}{dx}) \]

3. **Differentiate the Right Side Using the Product Rule:**
For \( xy \), we use the product rule:
\[ \frac{d}{dx}[xy] = x \frac{dy}{dx} + y \]

4. **Set the Derivatives Equal:**
Now we equate both sides:
\[ \sec(x+y) \tan(x+y) (1 + \frac{dy}{dx}) = x \frac{dy}{dx} + y \]

5. **Expand and Rearrange:**
Expanding the left side:
\[ \sec(x+y) \tan(x+y) + \sec(x+y) \tan(x+y) \frac{dy}{dx} = x \frac{dy}{dx} + y \]
Rearranging gives:
\[ \sec(x+y) \tan(x+y) \frac{dy}{dx} - x \frac{dy}{dx} = y - \sec(x+y) \tan(x+y) \]

6. **Factor Out \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} (\sec(x+y) \tan(x+y) - x) = y - \sec(x+y) \tan(x+y) \]

7. **Solve for \( \frac{dy}{dx} \):**
\[ \frac{dy}{dx} = \frac{y - \sec(x+y) \tan(x+y)}{\sec(x+y) \tan(x+y) - x} \]

### Final Answer:
\[ \frac{dy}{dx} = \frac{y - \sec(x+y) \tan(x+y)}{\sec(x+y) \tan(x+y) - x} \]