If `sqrt (x + y ) - sqrt (y -x) =c,` then ` (d ^(2)y )/( dx ^(2))` equals

Options

A. `(2)/(c^(2))`
B. `(-2)/(c ^(2))`
C. `(2)/(c )`
D. `(-2)/(c )`

Answer

To solve the problem, we start with the equation:

\[ \sqrt{x + y} - \sqrt{y - x} = c \]

### Step 1: Square both sides
We square both sides to eliminate the square roots:

\[ (\sqrt{x + y} - \sqrt{y - x})^2 = c^2 \]

Expanding the left side using the formula \((a - b)^2 = a^2 - 2ab + b^2\):

\[ (x + y) + (y - x) - 2\sqrt{(x + y)(y - x)} = c^2 \]

### Step 2: Simplify the equation
Now, we simplify the left side:

\[ y + y - 2\sqrt{(x + y)(y - x)} = c^2 \]

This simplifies to:

\[ 2y - 2\sqrt{(x + y)(y - x)} = c^2 \]

### Step 3: Rearrange the equation
Rearranging gives us:

\[ 2\sqrt{(x + y)(y - x)} = 2y - c^2 \]

Dividing both sides by 2:

\[ \sqrt{(x + y)(y - x)} = y - \frac{c^2}{2} \]

### Step 4: Square again
Now, we square both sides again:

\[ (x + y)(y - x) = \left(y - \frac{c^2}{2}\right)^2 \]

Expanding both sides:

\[ xy - x^2 + y^2 - xy = y^2 - c^2y + \frac{c^4}{4} \]

This simplifies to:

\[ -x^2 = -c^2y + \frac{c^4}{4} \]

### Step 5: Rearranging
Rearranging gives us:

\[ x^2 = c^2y - \frac{c^4}{4} \]

### Step 6: Differentiate with respect to x
Now we differentiate both sides with respect to \(x\):

\[ 2x = c^2 \frac{dy}{dx} \]

### Step 7: Solve for \(\frac{dy}{dx}\)
Rearranging gives:

\[ \frac{dy}{dx} = \frac{2x}{c^2} \]

### Step 8: Differentiate again to find \(\frac{d^2y}{dx^2}\)
Now we differentiate \(\frac{dy}{dx}\) again:

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2x}{c^2}\right) \]

Since \(c^2\) is a constant, we have:

\[ \frac{d^2y}{dx^2} = \frac{2}{c^2} \]

### Final Answer
Thus, the second derivative \(\frac{d^2y}{dx^2}\) is:

\[ \frac{d^2y}{dx^2} = \frac{2}{c^2} \]