find the derivative of y with respect to x.\n\n$y = \\frac { \\ln x } { 5 + 3 \\ln x }$\n\n$\\frac { d y } {…

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u = \ln x), (u^\prime=\frac{1}{x}), (v = 5 + 3\ln x), and (v^\prime=\frac{3}{x}).

Step2: Substitute into the quotient rule formula

[ \begin{align*} \frac{dy}{dx}&=\frac{\frac{1}{x}(5 + 3\ln x)-\ln x\cdot\frac{3}{x}}{(5 + 3\ln x)^{2}}\ &=\frac{\frac{5}{x}+\frac{3\ln x}{x}-\frac{3\ln x}{x}}{(5 + 3\ln x)^{2}} \end{align*} ]

Step3: Simplify the numerator

The (\frac{3\ln x}{x}-\frac{3\ln x}{x}) terms cancel out in the numerator. So we have (\frac{dy}{dx}=\frac{\frac{5}{x}}{(5 + 3\ln x)^{2}})

Step4: Final simplification

[ \frac{dy}{dx}=\frac{5}{x(5 + 3\ln x)^{2}} ]

Answer:

(\frac{5}{x(5 + 3\ln x)^{2}})