Calculate $\frac{d y}{d x}$. You need not expand your answer. \[ y=\frac{x^{2}+6 x-1}{x^{2}+5 x-1} \] \[ \frac{d y}{d x}= \]

Step 1 ：Given the function \(y=\frac{x^{2}+6 x-1}{x^{2}+5 x-1}\), we are asked to find the derivative of the function.

Step 2 ：We can use the quotient rule to find the derivative. The quotient rule states that the derivative of \(\frac{u}{v}\) is \(\frac{vu' - uv'}{v^2}\), where \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives.

Step 3 ：In this case, \(u = x^{2}+6 x-1\) and \(v = x^{2}+5 x-1\).

Step 4 ：We first need to find \(u'\) and \(v'\). The derivative of \(u\) is \(u' = 2x + 6\) and the derivative of \(v\) is \(v' = 2x + 5\).

Step 5 ：Now that we have \(u'\) and \(v'\), we can substitute these into the quotient rule formula to find \(\frac{dy}{dx}\).

Step 6 ：Substituting \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula, we get \(\frac{dy}{dx} = \frac{-(2x + 5)(x^{2} + 6x - 1) + (2x + 6)(x^{2} + 5x - 1)}{(x^{2} + 5x - 1)^{2}}\).

Step 7 ：\(\boxed{\frac{dy}{dx} = \frac{-(2x + 5)(x^{2} + 6x - 1) + (2x + 6)(x^{2} + 5x - 1)}{(x^{2} + 5x - 1)^{2}}}\) is the derivative of the function \(y=\frac{x^{2}+6 x-1}{x^{2}+5 x-1}\).