Differential equations and direction fields I am told that the equation
$\dfrac{dx}{dt} = f(t, x)$
means that for any point in the tx-plane (for which $f$ is defined) we can evaluate the gradient $\dfrac{dx}{dt}$ and represent this graphically by means of a small arrow representing the vector $\left(1, \dfrac{dx}{dt} \right)$.
However, I do not understand how we get $\left(1, \dfrac{dx}{dt} \right)$ from $\dfrac{dx}{dt} = f(t, x)$. 
Please explain the reasoning behind this and prove if it is true.
Thanks.

You are actually considering the direction field on the $xt$-plane give  by the system
\begin{align}
t' =&\ 1\\
x' =&\ f(x, t).
\end{align}
