Add $ \dfrac{1}{x} $ and $ \dfrac{1}{y} $ to get $ \dfrac{ \color{purple}{ x+y } }{ xy }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ y }$ and the second by $\color{blue}{ x }$.

$$ \begin{aligned} \frac{1}{x} + \frac{1}{y} & = \frac{ 1 \cdot \color{blue}{ y }}{ x \cdot \color{blue}{ y }} + \frac{ 1 \cdot \color{blue}{ x }}{ y \cdot \color{blue}{ x }} = \\[1ex] &=\frac{ \color{purple}{ y } }{ xy } + \frac{ \color{purple}{ x } }{ xy }=\frac{ \color{purple}{ x+y } }{ xy } \end{aligned} $$

Divide $1$ by $ \dfrac{x+y}{xy} $ to get $ \dfrac{ xy }{ x+y } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{ \frac{\color{blue}{x+y}}{\color{blue}{xy}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{xy}}{\color{blue}{x+y}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{xy}{x+y} \xlongequal{\text{Step 3}} \frac{ 1 \cdot xy }{ 1 \cdot \left( x+y \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ xy }{ x+y } \end{aligned} $$