1. The problem asks to interpret $\frac{dy}{dx}$ for the implicit function given by $\sin^{-1}(x + y) = \cos^{-1}(xy)$.\n\n2. $\frac{dy}{dx}$ represents the derivative of $y$ with respect to $x$, which measures how $y$ changes as $x$ changes. For implicit functions, we use implicit differentiation to find $\frac{dy}{dx}$.\n\n3. Differentiate both sides of the equation with respect to $x$. Recall the derivatives: $\frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1-u^2}} \frac{du}{dx}$ and $\frac{d}{dx} \cos^{-1}(v) = -\frac{1}{\sqrt{1-v^2}} \frac{dv}{dx}$.\n\n4. Let $u = x + y$ and $v = xy$. Then:\n$$\frac{d}{dx} \sin^{-1}(x + y) = \frac{1}{\sqrt{1-(x+y)^2}} (1 + \frac{dy}{dx})$$\n$$\frac{d}{dx} \cos^{-1}(xy) = -\frac{1}{\sqrt{1-(xy)^2}} \left(y + x \frac{dy}{dx}\right)$$\n\n5. Set derivatives equal:\n$$\frac{1}{\sqrt{1-(x+y)^2}} (1 + \frac{dy}{dx}) = -\frac{1}{\sqrt{1-(xy)^2}} \left(y + x \frac{dy}{dx}\right)$$\n\n6. Multiply both sides by denominators to clear fractions and group $\frac{dy}{dx}$ terms:\n$$\frac{1 + \frac{dy}{dx}}{\sqrt{1-(x+y)^2}} = -\frac{y + x \frac{dy}{dx}}{\sqrt{1-(xy)^2}}$$\n\n7. Multiply both sides by $\sqrt{1-(x+y)^2} \sqrt{1-(xy)^2}$:\n$$\sqrt{1-(xy)^2} (1 + \frac{dy}{dx}) = -\sqrt{1-(x+y)^2} (y + x \frac{dy}{dx})$$\n\n8. Expand:\n$$\sqrt{1-(xy)^2} + \sqrt{1-(xy)^2} \frac{dy}{dx} = -y \sqrt{1-(x+y)^2} - x \sqrt{1-(x+y)^2} \frac{dy}{dx}$$\n\n9. Group $\frac{dy}{dx}$ terms on one side:\n$$\sqrt{1-(xy)^2} \frac{dy}{dx} + x \sqrt{1-(x+y)^2} \frac{dy}{dx} = -y \sqrt{1-(x+y)^2} - \sqrt{1-(xy)^2}$$\n\n10. Factor $\frac{dy}{dx}$:\n$$\frac{dy}{dx} \left(\sqrt{1-(xy)^2} + x \sqrt{1-(x+y)^2}\right) = -y \sqrt{1-(x+y)^2} - \sqrt{1-(xy)^2}$$\n\n11. Solve for $\frac{dy}{dx}$:\n$$\frac{dy}{dx} = \frac{-y \sqrt{1-(x+y)^2} - \sqrt{1-(xy)^2}}{\sqrt{1-(xy)^2} + x \sqrt{1-(x+y)^2}}$$\n\n12. Interpretation: This derivative tells us the slope of the curve defined implicitly by the original equation at any point $(x,y)$. It shows how $y$ changes with $x$ considering the relationship between $x$ and $y$ given by the inverse sine and cosine functions.