Given a differentiable function of $x$, denoted by $f(x)$; is $f(x+a) - f(a) = f(x) + f'(a) x$ an identity?
For example, if $f(x)=x^2$, then it gives $(x+a)^2 - a^2 = x^2 + 2ax$, which is true.
So, if this identity is true, the derivative appears from a change of coordinate system (or translation)?
In general it is not true, take as an example the function $f(x)=c$, where $c\neq 0$ is a constant.