Rectilinear motion is motion in one direction. Imagine there was an origin, O, and an object was a distance, x, from the origin. Now, say the object traveled in a straight line to a distance $ x' $ from the origin during a time $ \Delta t $ such that the object traveled a total distance of $ \Delta x $. Then we can use this information to determine the object's velocity, and hence, its acceleration.
Using $ \Delta x $ and $ \Delta t $ from the example above, we can find the velocity as $$ v = \frac{dx}{dt} = \dot{x} $$. Notice that $ \delta $ was used instead of $ \Delta $ because $ \delta $ is used to quantify an infinitesimal amount of change while $ \Delta $ is used to quantify a change. If we were to also obtain the change in velocity over a change in time, $ \Delta t $, then we could also find the acceleration of the object as well. $$ a = \frac{d^2x}{dt^2} = \frac{d\dot{x}}{dt} = \ddot{x} $$
Now that Eq. (1) and Eq. (2) are defined, we can rearrange the two to derive an important equation that will be used throughout this course. We start with Eq. (1) to solve for $ dt $: $$ \frac{dx}{dt} = \dot{x} \longrightarrow dt = \frac{dx}{\dot{x}} $$ Then we can solve Eq. (2) for $ dt $: $$ \frac{d\dot{x}}{dt} = \ddot{x} \longrightarrow dt = \frac{d\dot{x}}{\ddot{x}} $$ Using Eq. (3) and Eq. (4), we can get: $$ \frac{dx}{\dot{x}} = \frac{d\dot{x}}{\ddot{x}} $$ $$ \Rightarrow \ddot{x} dx = \dot{x} d\dot{x} $$ This will come in handy much more as we go further into the course!