If $\sqrt{x}+\sqrt{y}=11$ and $y(4)=81$, find $y^{\prime}(4)$ by implicit differentiation.

Step 1 ：We are given the equation \(\sqrt{x} + \sqrt{y} = 11\) and the value of y at x=4, which is 81.

Step 2 ：We are asked to find the derivative of y with respect to x, denoted as \(y'(x)\) or \(\frac{dy}{dx}\).

Step 3 ：Since y is not explicitly defined as a function of x, we need to use implicit differentiation.

Step 4 ：Differentiating both sides of the equation with respect to x, we get \(\frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot y'(x) = 0\).

Step 5 ：Solving this equation for \(y'(x)\), we get \(y'(x) = -\frac{\sqrt{y}}{\sqrt{x}}\).

Step 6 ：Substituting the given values x=4 and y=81 into this equation, we find that \(y'(4) = -\frac{9}{2}\).

Step 7 ：Final Answer: \(\boxed{-\frac{9}{2}}\)