Question

find the derivative of y with respect to x.
y = \frac{\ln x}{2 + 3\ln x}
\frac{dy}{dx}=\square

Explanation:

Step1: Apply quotient - rule

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

Step2: Substitute into quotient - rule formula

$y^\prime=\frac{\frac{1}{x}(2 + 3\ln x)-\ln x\cdot\frac{3}{x}}{(2 + 3\ln x)^{2}}$.

Step3: Simplify the numerator

$\frac{1}{x}(2 + 3\ln x)-\ln x\cdot\frac{3}{x}=\frac{2+3\ln x - 3\ln x}{x}=\frac{2}{x}$.

Answer:

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