Question

find the derivative of y with respect to x.

$y = \frac { \ln x } { 5 + 3 \ln x }$

$\frac { d y } { d x } = \square$

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

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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:

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