[Math] Taylor expansion of a variable with small displacement

Question

I got a question concerning the Taylor expansion of a function
$ F(\vec{x},t+\Delta t)$,
where $\Delta t$ is a small displacement with $\Delta t \rightarrow 0$.

In one of my books it is stated that the expansion looks like
$$ F(\vec{x},t+\Delta t) \approx F(\vec{x},t)+\Delta t\frac{\partial F(\vec{x},t)}{\partial t} $$

But I can't really get my head around on how to get there with a formal approach.
Can anybody give me a hint or explanation?

Answer 1

Hint: For every fixed $x$, the map $g(t) = F(x,t)$ is a function of one variable such that $g'(t) = \partial F/\partial t(x,t)$. What is the Taylor expansion of $g$?