Derivative of f(x) = 2x ln x and g(x) = (5 − 2 ln(x-1))/x

The product rule states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.
Let $u(x) = 2x$ and $v(x) = \ln x$.

2) For $g(x) = \frac{5 - 2 \ln(x-1)}{x}$

The quotient rule states that if $g(x) = \frac{u(x)}{v(x)}$, then $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$.
Let $u(x) = 5 - 2 \ln(x-1)$ and $v(x) = x$.

Using the chain rule for $\ln(x-1)$, where the derivative of $\ln(w)$ is $\frac{1}{w}$ and the derivative of $x-1$ is 1:

To express the numerator as a single fraction: