Answer is: option2

Solution:

We are given the equation \( y = x + \cos(xy) \) and asked to find the slope of the normal line at the point \( (0, 1) \).

Step 1: Differentiate the equation implicitly

Differentiating both sides with respect to \( x \):

\[ \frac{d}{dx}(y) = \frac{d}{dx}(x + \cos(xy)) \]

Applying the chain rule:

\[ \frac{dy}{dx} = 1 - \sin(xy) \cdot (y + x \frac{dy}{dx}) \]

Step 2: Solve for \( \frac{dy}{dx} \)

Substituting the point \( (0, 1) \) into the equation:

\[ \frac{dy}{dx} = 1 - \sin(0 \cdot 1) \cdot (1 + 0 \cdot \frac{dy}{dx}) = 1 \]

Step 3: Find the slope of the normal line

The slope of the normal line is the negative reciprocal of the slope of the tangent line:

\[ m_{\text{normal}} = -\frac{1}{1} = -1 \]

Final Answer

The slope of the normal line is \( \boxed{-1} \).