Taylor expansion vs expansion in physics

Question

In physics I often see expansions of the form $\phi(x+a)=\phi(x)+a\phi'(x)$ for small a. How does this coincide with the usual Taylor expansion $\phi(x-a)=\phi(a)+\phi'(a)(x-a) +...$ ?

Answer 1

The derivative $f'(x)$ of a function $f(x)$ can be defined as $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$ Thus for small $h$, $$f'(x)\approx \frac{f(x+h)-f(x)}{h}$$ Or, $$f(x+h)\approx f(x)+hf'(x)$$

Answer 2

The first is an expansion around $x = 0$, for small $a$ whilst the second one is around $x = a$.

Answer 3

The first is an expansion around $x$, with a small, and it is a not rigorous way to write

$\phi(x+a)\approx \phi(x)+a\phi'(x)$

or

$\phi(x+a)\sim \phi(x)+a\phi'(x)$

the second is the Taylor’s expansion around x=a.