Use logarithmic differentiation to find $y^{\prime}$.
\[
y=\frac{\sqrt{7-5 x}\left(x^{2}+1\right)^{2}}{x^{2}+5 x+7}
\]
\[
y^{\prime}=
\]
\(\boxed{y' = 4x\sqrt{7 - 5x}\frac{(x^2 + 1)}{(x^2 + 5x + 7)} + \sqrt{7 - 5x}(-2x - 5)\frac{(x^2 + 1)^2}{(x^2 + 5x + 7)^2} - 5\frac{(x^2 + 1)^2}{2\sqrt{7 - 5x}(x^2 + 5x + 7)}}\)
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 :\(\boxed{y' = 4x\sqrt{7 - 5x}\frac{(x^2 + 1)}{(x^2 + 5x + 7)} + \sqrt{7 - 5x}(-2x - 5)\frac{(x^2 + 1)^2}{(x^2 + 5x + 7)^2} - 5\frac{(x^2 + 1)^2}{2\sqrt{7 - 5x}(x^2 + 5x + 7)}}\)