For the curve `sqrt(x) + sqrt(y) = 1, (dy)/(dx)` at `(1/4,1/4)` is `"………."`.

Explanation

To find \(\frac{dy}{dx}\) for the curve \(\sqrt{x} + \sqrt{y} = 1\) at the point \((\frac{1}{4}, \frac{1}{4})\), we will follow these steps:

### Step 1: Differentiate the equation implicitly
We start with the equation:
\[ \sqrt{x} + \sqrt{y} = 1 \]
We will differentiate both sides with respect to \(x\).

### Step 2: Apply the chain rule
Differentiating \(\sqrt{x}\) gives:
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
For \(\sqrt{y}\), we apply the chain rule:
\[ \frac{d}{dx}(\sqrt{y}) = \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} \]
Thus, differentiating the entire equation gives:
\[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = 0 \]

### Step 3: Solve for \(\frac{dy}{dx}\)
Rearranging the equation to isolate \(\frac{dy}{dx}\):
\[ \frac{1}{2\sqrt{y}} \cdot \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \]
Multiplying both sides by \(2\sqrt{y}\) to eliminate the fraction:
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]

### Step 4: Substitute the point \((\frac{1}{4}, \frac{1}{4})\)
Now we substitute \(x = \frac{1}{4}\) and \(y = \frac{1}{4}\) into the derivative:
\[ \frac{dy}{dx} = -\frac{\sqrt{\frac{1}{4}}}{\sqrt{\frac{1}{4}}} \]
Calculating the square roots:
\[ \frac{dy}{dx} = -\frac{\frac{1}{2}}{\frac{1}{2}} = -1 \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at the point \((\frac{1}{4}, \frac{1}{4})\) is:
\[ \frac{dy}{dx} = -1 \]

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