For the curve `sqrt(x)+sqrt(y)=1` , `(dy)/(dx)` at `(1//4,\ 1//4)` is `1//2` (b) 1 (c) -1 (d) 2

Explanation

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

### Step 1: Differentiate the equation implicitly
We start with the equation:
\[ \sqrt{x} + \sqrt{y} = 1 \]
Differentiating both sides with respect to \(x\):
\[ \frac{d}{dx}(\sqrt{x}) + \frac{d}{dx}(\sqrt{y}) = \frac{d}{dx}(1) \]

### Step 2: Apply the differentiation rules
Using the differentiation rule for square roots, we have:
\[ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \frac{dy}{dx} = 0 \]

### Step 3: Rearrange the equation
Now, we can rearrange this equation to solve for \(\frac{dy}{dx}\):
\[ \frac{1}{2\sqrt{y}} \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \]
Multiplying both sides by \(2\sqrt{y}\) gives:
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]

### Step 4: Substitute the point \(\left(\frac{1}{4}, \frac{1}{4}\right)\)
Now we substitute \(x = \frac{1}{4}\) and \(y = \frac{1}{4}\) into the equation for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = -\frac{\sqrt{\frac{1}{4}}}{\sqrt{\frac{1}{4}}} \]
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 \(\left(\frac{1}{4}, \frac{1}{4}\right)\) is:
\[ \frac{dy}{dx} = -1 \]

### Conclusion
The correct option is (c) -1.