If `sqrt(x+y) +sqrt(y-x)=5,` then ` (d^(2)y)/(dx ^(2))=`

Options

A. `2/5`
B. `4/25`
C. `2/25`
D. `1/25`

Answer

To solve the problem, we need to find the second derivative of \( y \) with respect to \( x \) given the equation:

\[ \sqrt{x+y} + \sqrt{y-x} = 5 \]

### Step 1: Square both sides

First, we square both sides of the equation to eliminate the square roots:

\[ (\sqrt{x+y} + \sqrt{y-x})^2 = 5^2 \]

This expands to:

\[ x+y + y-x + 2\sqrt{(x+y)(y-x)} = 25 \]

### Step 2: Simplify the equation

Now, simplify the left side:

\[ 2y + 2\sqrt{(x+y)(y-x)} = 25 \]

### Step 3: Isolate the square root

Next, isolate the square root term:

\[ 2\sqrt{(x+y)(y-x)} = 25 - 2y \]

### Step 4: Square both sides again

Now, square both sides again to eliminate the square root:

\[ 4(x+y)(y-x) = (25 - 2y)^2 \]

### Step 5: Expand both sides

Expanding both sides gives:

\[ 4(xy - x^2 + y^2 - xy) = 625 - 100y + 4y^2 \]

This simplifies to:

\[ 4(y^2 - x^2) = 625 - 100y + 4y^2 \]

### Step 6: Rearrange the equation

Rearranging the equation leads to:

\[ 0 = 625 - 100y + 4y^2 - 4y^2 + 4x^2 \]

This simplifies to:

\[ 100y = 625 + 4x^2 \]

### Step 7: Differentiate both sides

Now, differentiate both sides with respect to \( x \):

\[ 100 \frac{dy}{dx} = 8x \]

### Step 8: Solve for \( \frac{dy}{dx} \)

From this, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{8x}{100} = \frac{2x}{25} \]

### Step 9: Differentiate again to find \( \frac{d^2y}{dx^2} \)

Now, differentiate \( \frac{dy}{dx} \) again to find \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2x}{25}\right) = \frac{2}{25} \]

### Final Answer

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

\[ \frac{d^2y}{dx^2} = \frac{2}{25} \]