The given equation is: \[ x^y + y^x = a^b \quad \text{(constant)} \] Differentiate both sides with respect to \( x \) using implicit differentiation. Derivative of \( x^y \) with respect to \( x \) is: \[ \frac{d}{dx}(x^y) = x^y \left( \frac{y}{x} + \ln x \cdot \frac{dy}{dx} \right) \] Derivative of \( y^x \) with respect to \( x \) is: \[ \frac{d}{dx}(y^x) = y^x (\ln y) + x y^x \cdot \frac{1}{y} \cdot \frac{dy}{dx} \] Applying the sum rule for differentiation: \[ \frac{d}{dx}(x^y) + \frac{d}{dx}(y^x) = 0 \] Substituting the derivatives: \[ x^y \left( \frac{y}{x} + \ln x \cdot \frac{dy}{dx} \right) + y^x \left( \ln y + \frac{x}{y} \cdot \frac{dy}{dx} \right) = 0 \] Substitute \( x = 1 \) and \( y = 2 \): \[ x^y = 1^2 = 1, \quad y^x = 2^1 = 2, \quad \ln(1) = 0, \quad \ln(2) \approx 0.693 \] \[ 1 \cdot \left( \frac{2}{1} + 0 \cdot \frac{dy}{dx} \right) + 2 \cdot \left( \ln 2 + \frac{1}{2} \cdot \frac{dy}{dx} \right) = 0 \] Simplifying the equation: \[ 2 + 2 \left( \ln 2 + \frac{1}{2} \cdot \frac{dy}{dx} \right) = 0 \] \[ 2 + 2 \ln 2 + \frac{dy}{dx} = 0 \] Solving for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = -2 - 2 \ln 2 \] Using the approximation \( \ln 2 \approx 0.693 \): \[ \frac{dy}{dx} \approx -2 - 2(0.693) = -2 - 1.386 = -3.386 \] A more direct implicit differentiation approach yields the same result. Given \( x^y + y^x = \text{constant} \), differentiating both sides gives \( \frac{d}{dx}(x^y) + \frac{d}{dx}(y^x) = 0 \). Using logarithmic differentiation, we find \( \frac{d}{dx}(x^y) = x^y \left( \ln x \cdot \frac{dy}{dx} + \frac{y}{x} \right) \) and \( \frac{d}{dx}(y^x) = y^x \left( \ln y + x \cdot \frac{1}{y} \cdot \frac{dy}{dx} \right) \). At \( x = 1, y = 2 \): - \( x^y = 1 \), \( y^x = 2 \) - \( \ln x = 0 \), \( \ln y = \ln 2 \) Substituting these values: \[ 1 \cdot \left( 0 + \frac{2}{1} \right) + 2 \cdot \left( \ln 2 + \frac{1}{2} \cdot \frac{dy}{dx} \right) = 0 \] \[ 2 + 2 \ln 2 + \frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = - (2 + 2 \ln 2) \approx - (2 + 1.386) = -3.386 \] The calculated value is approximately \( -3.386 \). This value does not directly match any provided options. If forced to choose the closest option in a multiple-choice context, and assuming potential approximation inaccuracies or intended simpler values, \( -1 \) might be considered, though it's not a precise match. None of the options precisely align with the calculated result.