Constant product market maker expansion for LW3

constant-product-market-maker-lw3.tex

\documentclass{article}
\usepackage[utf8]{inputenc}
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\begin{document}

\begin{align*}
    x y& = k\\
    (x + \Delta x) (y - \Delta y)& = k\\
    (x + \Delta x) (y - \Delta y)& = x y
\end{align*}

Flip it, then:

\begin{align*}
    x y& = (x + \Delta x) (y - \Delta y)\\
    x y& = y(x + \Delta x) - \Delta y(x + \Delta x)\\
    x y& = x y + y \Delta x - \Delta y x - \Delta y \Delta x\\
    x y - x y& =  y \Delta x - \Delta y x - \Delta y \Delta x\\
    0& =  y \Delta x - \Delta y x - \Delta y \Delta x\\
    \Delta y x& =  y \Delta x - \Delta y \Delta x\\
    \Delta y x + \Delta y \Delta x& =  y \Delta x\\
    \Delta y (x + \Delta x)& =  y \Delta x\\
    \Delta y& =  \frac{y \Delta x}{(x + \Delta x)}\\
\end{align*}

\end{document}