Differentiation… from first principles

‘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:

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