Answer
$\frac{1}{y}\cdot\frac{y}{x}$
Step-by-step explanation
Problem to solve:
$\frac{d}{dx}\left(\ln\left(\frac{x}{y}\right)\right)$
1
The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{\frac{x}{y}}\cdot\frac{d}{dx}\left(\frac{x}{y}\right)$
2
Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{1}{\frac{x}{y}}\cdot\frac{y\frac{d}{dx}\left(x\right)-x\frac{d}{dx}\left(y\right)}{y^2}$
Answer
$\frac{1}{y}\cdot\frac{y}{x}$