‘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, d*s*/d*t*, 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: