1. **State the problem:** We are given the implicit equation $$\psi_5(x,y) = x \cdot (xy) - 1 = 0$$ and asked to find the derivative $\frac{dy}{dx}$. 2. **Rewrite the equation:** Simplify the expression: $$x \cdot (xy) - 1 = x^2 y - 1 = 0$$ 3. **Implicit differentiation:** Differentiate both sides with respect to $x$: $$\frac{d}{dx}(x^2 y) - \frac{d}{dx}(1) = 0$$ 4. **Apply product rule to $x^2 y$:** $$\frac{d}{dx}(x^2 y) = 2x y + x^2 \frac{dy}{dx}$$ 5. **Substitute back:** $$2x y + x^2 \frac{dy}{dx} - 0 = 0$$ 6. **Solve for $\frac{dy}{dx}$:** $$x^2 \frac{dy}{dx} = -2x y$$ 7. **Cancel common factor $x$ (assuming $x \neq 0$):** $$\cancel{x} x \frac{dy}{dx} = -2 \cancel{x} y$$ $$x \frac{dy}{dx} = -2 y$$ 8. **Divide both sides by $x$ to isolate $\frac{dy}{dx}$:** $$\cancel{x} \frac{dy}{dx} = -2 y \frac{1}{\cancel{x}}$$ $$\frac{dy}{dx} = -\frac{2 y}{x}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = -\frac{2 y}{x}}$$