Quotient rule with logarithmic and trigonometric functions

Identify $f(x) = \ln(x)$ and $g(x) = \sin(x)$, so $f'(x) = \dfrac{1}{x}$ and $g'(x) = \cos(x)$. Apply the quotient rule: $h'(x) = \dfrac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$.

Substituting gives $h'(x) = \dfrac{\frac{1}{x}\sin(x) - \ln(x)\cos(x)}{\sin^2(x)}$. This can be left as is or rewritten with a common denominator inside the numerator as $\dfrac{\sin(x) - x\ln(x)\cos(x)}{x\sin^2(x)}$.