‘Velocity’, v, is defined as distance travelled, s, over a time interval, t:
But what if the velocity is changing constantly? In this case, we can consider the velocity at any time, by making the time interval as short as possible:
The quantity, ds/dt, is called the ‘derivative of s with respect to t’.
Now, consider the motion of a point described by the formula:
We want to know the velocity at any time, t. Well, the distance travelled after ‘t plus a short time interval’ will be ‘s plus a short distance’. Let’s substitute these into the above formula and do some calculations along the way:
Let’s now substitute s back in from our original equation:
Dividing by the short time interval gives us:
Making the time interval infinitesimally small:
So, for…
… the derivative of s with respect to t is:
