If `sqrt(y+x)+sqrt(y-x)=c,` where `cne0`, then `(dy)/(dx)` has the value equal to

Options

A. `(2x)/(e^(2))`
B. `(x)/(y+sqrt(y^(2)-x^(2)))`
C. `(y-sqrt(y^(2)-x^(2)))/(x)`
D. `(c^(2))/(2y)`

Answer

To find \(\frac{dy}{dx}\) given the equation \(\sqrt{y+x} + \sqrt{y-x} = c\), where \(c \neq 0\), we will differentiate both sides with respect to \(x\).

### Step-by-Step Solution:

1. **Differentiate Both Sides**:
\[ \frac{d}{dx}(\sqrt{y+x}) + \frac{d}{dx}(\sqrt{y-x}) = \frac{d}{dx}(c) \]
Since \(c\) is a constant, its derivative is 0:
\[ \frac{d}{dx}(\sqrt{y+x}) + \frac{d}{dx}(\sqrt{y-x}) = 0 \]

2. **Apply the Chain Rule**:
For \(\sqrt{y+x}\):
\[ \frac{1}{2\sqrt{y+x}} \left( \frac{dy}{dx} + 1 \right) \]
For \(\sqrt{y-x}\):
\[ \frac{1}{2\sqrt{y-x}} \left( \frac{dy}{dx} - 1 \right) \]
Therefore, we have:
\[ \frac{1}{2\sqrt{y+x}} \left( \frac{dy}{dx} + 1 \right) + \frac{1}{2\sqrt{y-x}} \left( \frac{dy}{dx} - 1 \right) = 0 \]

3. **Combine the Terms**:
\[ \frac{1}{2\sqrt{y+x}} \left( \frac{dy}{dx} + 1 \right) + \frac{1}{2\sqrt{y-x}} \left( \frac{dy}{dx} - 1 \right) = 0 \]
Multiplying through by \(2\) to eliminate the fraction:
\[ \frac{1}{\sqrt{y+x}} \left( \frac{dy}{dx} + 1 \right) + \frac{1}{\sqrt{y-x}} \left( \frac{dy}{dx} - 1 \right) = 0 \]

4. **Distribute and Rearrange**:
\[ \frac{1}{\sqrt{y+x}} \frac{dy}{dx} + \frac{1}{\sqrt{y+x}} + \frac{1}{\sqrt{y-x}} \frac{dy}{dx} - \frac{1}{\sqrt{y-x}} = 0 \]
Combine the \(\frac{dy}{dx}\) terms:
\[ \left( \frac{1}{\sqrt{y+x}} + \frac{1}{\sqrt{y-x}} \right) \frac{dy}{dx} + \left( \frac{1}{\sqrt{y+x}} - \frac{1}{\sqrt{y-x}} \right) = 0 \]

5. **Isolate \(\frac{dy}{dx}\)**:
\[ \left( \frac{1}{\sqrt{y+x}} + \frac{1}{\sqrt{y-x}} \right) \frac{dy}{dx} = -\left( \frac{1}{\sqrt{y+x}} - \frac{1}{\sqrt{y-x}} \right) \]
Thus,
\[ \frac{dy}{dx} = -\frac{\frac{1}{\sqrt{y+x}} - \frac{1}{\sqrt{y-x}}}{\frac{1}{\sqrt{y+x}} + \frac{1}{\sqrt{y-x}}} \]

6. **Simplify the Expression**:
To simplify, multiply numerator and denominator by \(\sqrt{y+x} \cdot \sqrt{y-x}\):
\[ \frac{dy}{dx} = -\frac{\sqrt{y-x} - \sqrt{y+x}}{\sqrt{y-x} + \sqrt{y+x}} \]

### Final Answer:
\[ \frac{dy}{dx} = -\frac{\sqrt{y-x} - \sqrt{y+x}}{\sqrt{y-x} + \sqrt{y+x}} \]