Can please someone explain (or provide a reference) which explains the following:

Assuming we have an arbitrary function f(x). If we increase x by delta x, the new function at x + dx is:

f(x+dx)= f(x) + d/dx f(x) dx

So far I have noticed it works, if I use generic formulas (e.g. f(x)=ax and f(x+dx)=a(x+dx)), but I don't fully understand this.

Any help will be much appreciated