$\left(\frac{x+3}{(x+1)}\right)^{\prime}$

Step 1 ：Given function: \(\frac{x+3}{x+1}\)

Step 2 ：Find the derivative using the quotient rule: \(\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}\)

Step 3 ：Compute the derivatives: \(f'(x) = 1\) and \(g'(x) = 1\)

Step 4 ：Apply the quotient rule: \(\frac{1(x+1) - (x+3)(1)}{(x+1)^2}\)

Step 5 ：Simplify the expression: \(\frac{-2}{(x+1)^2}\)

Step 6 ：\(\boxed{\frac{-2}{(x+1)^2}}\)