How to build Automatic Differentiation script with just python

let’s say we have

\[f(x,y) = x^2 + xy + 1\]

and we can rewrite it as

\[f(x,y) = (x \cdot x) + (x \cdot y + 1)\] \[f(x,y) = Sum[(x \cdot x), (x \cdot y + 1)]\] \[f(x,y) = Sum[(x \cdot x), Sum[x \cdot y, 1]]\] \[f(x,y) = Sum[Mul[x, x], Sum[Mul[x, y], 1]]\]

or

\[f(x,y) = Sum[Pow[x, 2], Sum[Mul[x, y], 1]]\] \[\frac{d}{d\psi} Sum[a, b] = \frac{da}{d\psi} + \frac{db}{d\psi}\] \[\frac{d}{d\psi} Mul[a, b] = a\frac{db}{d\psi} + b\frac{da}{d\psi}\] \[\frac{d}{d\psi} Pow[base, constant] = constant \cdot base^{constant - 1}\] \[\frac{da}{d\psi}= \begin{cases} 1 & \text{if } \psi = a\\ 0 & \text{if } \psi \neq a \end{cases}\] \[\frac{d (constant)}{d\psi} = 0\]