Given the perturbation $x\rightarrow x+\delta x$, how can the function $$ \tanh[c(x-x_0)], $$ $c$ and $x_0$ constant, be perturbed? I'd like to have a form like $$ f(x)\rightarrow f(x)+g(\delta x) $$ such that I have the different orders separated. But I am not certain how I can split $\tanh[c(x+\delta x-x_0)]$.
Is there a standard procedure? A Taylor expansion would not lead to separated terms, because all the $\delta x$ terms would have a factor depending on $x$ as well, e.g. in a somewhat simpler example:
$$ \exp(cx)\rightarrow\exp(cx)\exp(c\delta x)\approx\exp(cx)(1+c\delta x) $$
for $\delta x$ small.
EDIT As pointed out in the comments (thx!) a Taylor Series around $\delta x=0$ is the way to go and the result even have to depend on $x$, so therefore I expect that $$ \tanh[c(x-x_0)]\rightarrow\tanh[c(x+\delta x-x_0)]\approx\tanh[c(x-x_0)]+\frac{c\delta x}{\cosh^2[c(x-x_0)]} $$ is the way to go.