1. **State the problem:** Find the derivative $\frac{dy}{dx}$ for the equation $$\sqrt{x} + \sqrt{y} = 4.$$\n\n2. **Rewrite the equation:** Recall that $\sqrt{x} = x^{\frac{1}{2}}$ and $\sqrt{y} = y^{\frac{1}{2}}$, so the equation is $$x^{\frac{1}{2}} + y^{\frac{1}{2}} = 4.$$\n\n3. **Differentiate both sides with respect to $x$:** Using implicit differentiation, we differentiate term-by-term. For $x^{\frac{1}{2}}$, the derivative is $$\frac{d}{dx} x^{\frac{1}{2}} = \frac{1}{2} x^{-\frac{1}{2}}.$$\nFor $y^{\frac{1}{2}}$, since $y$ is a function of $x$, use the chain rule: $$\frac{d}{dx} y^{\frac{1}{2}} = \frac{1}{2} y^{-\frac{1}{2}} \frac{dy}{dx}.$$\nThe derivative of the constant 4 is 0.\n\n4. **Write the differentiated equation:** $$\frac{1}{2} x^{-\frac{1}{2}} + \frac{1}{2} y^{-\frac{1}{2}} \frac{dy}{dx} = 0.$$\n\n5. **Solve for $\frac{dy}{dx}$:**\n\n$$\frac{1}{2} y^{-\frac{1}{2}} \frac{dy}{dx} = - \frac{1}{2} x^{-\frac{1}{2}}$$\nMultiply both sides by 2 to clear denominators:\n$$y^{-\frac{1}{2}} \frac{dy}{dx} = - x^{-\frac{1}{2}}$$\nDivide both sides by $y^{-\frac{1}{2}}$:\n$$\frac{dy}{dx} = - x^{-\frac{1}{2}} \cdot y^{\frac{1}{2}} = - \frac{\sqrt{y}}{\sqrt{x}}.$$\n\n6. **Final answer:** $$\boxed{\frac{dy}{dx} = - \frac{\sqrt{y}}{\sqrt{x}}}.$$\n\nThis means the rate of change of $y$ with respect to $x$ depends on the ratio of the square roots of $y$ and $x$, with a negative sign indicating an inverse relationship.