Example of a quadratic function First a simple example in a physics context: the quadratic relation $s=a\cdot t^2$ between the distance $s$ travelled in a time $t$ by an object that is at time $t=0$ is at rest and thereafter undergoes a uniformly accelerated rectilinear motion with an acceleration $a$, which is constant in this example. The distance ${\vartriangle}s(t_1)$ between time $t=t_1$ and $t=t_1+{\vartriangle}t$ can be calculate in an exact way:

$${\vartriangle}s(t_1)=a(t_1+{\vartriangle}t)^2 -at_1^2=2at_1{\vartriangle}t+a({\vartriangle}t)^2\tiny.$$

If the time duration ${\vartriangle}t$ is short, then$({\vartriangle}t)^2$ is much smaller than ${\vartriangle}t$ and the last term is negligibly small in the above equation. so:

$${\vartriangle}s(t_1)\approx 2at_1{\vartriangle}t,\quad \text{for small }{\vartriangle}t\tiny.$$

The quantity $s$ is not only the distance traveled, but represent at the same time the position of the object at a certain time. Thus, the last equation can also be considered as an approximation for the change of position ${\vartriangle}s(t_1)$ on a time interval $[t_1,t_1+{\vartriangle}t]$ and this approximation is better when the change of time ${\vartriangle}t$ is smaller. Note, we are discussing here a relationship between changes in two variables (position and time). In order to indicate that the changes are negligibly small (infinitesimal'), one uses the symbols $ds$ and $dt$, and then the approximation becomes an equality

$$ds(t_1)=2a\cdot t_1\cdot dt$$ Because both $dt$ as $ds$ are understood as changes, that is, as difference between final and initial value, they are called differentials. The ratio of these differentials, the differential quotient $\displaystyle\frac{ds}{dt}$, is the slope of the tangent line at a point $(t,s(t))$ , that is, is equal to the derivative of $s$ in $t.$