Use logarithmic differentiation to find $y^{\prime}$. \[ y=\frac{\sqrt{7-5 x}\left(x^{2}+1\right)^{2}}{x^{2}+5 x+7} \] \[ y^{\prime}= \]

Step 1 ：Given the function \(y = \frac{\sqrt{7 - 5x}(x^{2} + 1)^{2}}{x^{2} + 5x + 7}\)

Step 2 ：Take the natural logarithm of both sides to simplify the differentiation process: \(\ln y = \ln \left(\frac{\sqrt{7 - 5x}(x^{2} + 1)^{2}}{x^{2} + 5x + 7}\right)\)

Step 3 ：Use the properties of logarithms to simplify the equation further

Step 4 ：Differentiate both sides of the equation with respect to x

Step 5 ：\(\frac{dy}{dx} = \frac{d}{dx} \left(\ln y\right)\)

Step 6 ：\(\frac{dy}{dx} = \frac{1}{y} \frac{dy}{dx}\)

Step 7 ：Solve for \(y'\) by multiplying both sides of the equation by y

Step 8 ：\(y' = 4x\sqrt{7 - 5x}\frac{(x^2 + 1)}{(x^2 + 5x + 7)} + \sqrt{7 - 5x}(-2x - 5)\frac{(x^2 + 1)^2}{(x^2 + 5x + 7)^2} - 5\frac{(x^2 + 1)^2}{2\sqrt{7 - 5x}(x^2 + 5x + 7)}\)

Step 9 ：\(\boxed{y' = 4x\sqrt{7 - 5x}\frac{(x^2 + 1)}{(x^2 + 5x + 7)} + \sqrt{7 - 5x}(-2x - 5)\frac{(x^2 + 1)^2}{(x^2 + 5x + 7)^2} - 5\frac{(x^2 + 1)^2}{2\sqrt{7 - 5x}(x^2 + 5x + 7)}}\)