Where does this equation for stationary points come from?

[Alexander350]

In the Classical Mechanics volume of The Theoretical Minimum, he writes a shorthand equation for a small change in a function. Please could someone explain exactly what it means and where it comes from?

 

[S.G. Janssens]

If you want to have it explained

exactly

then it requires a bit of calculus.

Namely, it is a theorem that if $A$ is a function depending on variables $x_1,\ldots,x_n$ and all partial derivatives $\frac{\partial A}{\partial x_i}$ exist as continuous functions, then the total derivative of $A$ is given by $\nabla A = (\frac{\partial A}{\partial x_1},\ldots,\frac{\partial A}{\partial x_n})$. This implies that the change
$$
A(\mathbf{x} + \Delta{\mathbf{x}}) - A(\mathbf{x}) = \nabla{A} \cdot \Delta{\mathbf{x}} + O(\|\Delta{x}\|^2) \qquad (\ast)
$$
where $\cdot$ denotes the inner product, so
$$
\nabla{A} \cdot \Delta{\mathbf{x}} = \sum_i{\frac{\partial A}{\partial x_i}\Delta x_i}
$$
and $O(\|\Delta{x}\|^2)$ are terms of at least quadratic order. I believe that physicists then argue that as $\|\Delta{x}\|$ becomes "infinitesimally small", these quadratic terms can be neglected and what is left in $(\ast)$ is denoted $\delta A$.