If `sqrt(x)+sqrt(y)=sqrt(a)," then "(dy)/(dx)=`

Options

A. `sqrt((x)/(y))`
B. `-(y)/(x)`
C. `-sqrt((y)/(x)`
D. `-sqrt(a)`

Answer

To solve the problem where \( \sqrt{x} + \sqrt{y} = \sqrt{a} \) and we need to find \( \frac{dy}{dx} \), we will follow these steps:

### Step 1: Rearrange the equation
Starting from the equation:
\[ \sqrt{x} + \sqrt{y} = \sqrt{a} \]
we can isolate \( \sqrt{y} \):
\[ \sqrt{y} = \sqrt{a} - \sqrt{x} \]

### Step 2: Differentiate both sides with respect to \( x \)
Now we will differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(\sqrt{y}) = \frac{d}{dx}(\sqrt{a} - \sqrt{x}) \]

### Step 3: Apply the chain rule
Using the chain rule on the left side and the differentiation rules on the right side:
\[ \frac{1}{2\sqrt{y}} \frac{dy}{dx} = 0 - \frac{1}{2\sqrt{x}} \]

### Step 4: Simplify the equation
This simplifies to:
\[ \frac{1}{2\sqrt{y}} \frac{dy}{dx} = -\frac{1}{2\sqrt{x}} \]

### Step 5: Solve for \( \frac{dy}{dx} \)
Now, we can solve for \( \frac{dy}{dx} \) by multiplying both sides by \( 2\sqrt{y} \):
\[ \frac{dy}{dx} = -\frac{2\sqrt{y}}{2\sqrt{x}} \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]

### Final Answer
Thus, the final result is:
\[ \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} \]

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