Find $\frac{d^{2} y}{d x^{2}}$. \[ y=\frac{x}{x+1} \] A. $(x+1)^{-2}$ B. $-2(x+1)^{-3}$ c. $-2(x+1)^{-2}$ D. $(x+1)^{-3}$

Step 1 ：We are given the function \(y=\frac{x}{x+1}\) and asked to find the second derivative, \(\frac{d^{2} y}{d x^{2}}\).

Step 2 ：First, we find the first derivative of the function. Using the quotient rule, we get \(y' = -\frac{x}{(x + 1)^2} + \frac{1}{x + 1}\).

Step 3 ：Next, we find the second derivative by differentiating the first derivative. This gives us \(y'' = \frac{2x}{(x + 1)^3} - \frac{2}{(x + 1)^2}\).

Step 4 ：However, none of the given options match this result. Let's simplify the expression for the second derivative.

Step 5 ：After simplifying, we find that \(y'' = -\frac{2}{(x + 1)^3}\).

Step 6 ：\(\boxed{-2(x+1)^{-3}}\) is the correct answer, which matches option B.