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# Derivatives

On a trip, the car speedometer shows the speed right now. Instantly. Instantaneously.

• If your car could measure the time for every 100 metres, it could calculate and show the average over 100 metres at a time. Not an instantaneous speed.
• Could it measure the time over each metre, the average would be much more useful. The speed probably didn’t change much over just one metre, so the average speed would be closer to the actual, instantaneous speed at each moment during that metre.
• Measuring the time over smaller and smaller distances, until almost ‘infinitely’ small, gives averages over smaller and smaller parts of the route, which become closer and closer to the actual instantaneous speed.

When distance becomes insignificantly tiny, ‘infinitely’ small or infinitesimal – so tiny that we are basically talking about just a point – then the average is the instantaneous speed.1 When that is the case, let’s change the ‘difference’ symbol from $\Delta$ to simply a $\mathrm d$.2Distance divided by duration in a single moment’ is then:

$$v=\frac{\mathrm ds}{\mathrm dt}$$

We might say that speed $v$ is derived from the distance $s$. $v$ is ‘the derivative of $s$ to $t$’.

This method of reducing values to become very, very tiny, until we reach something basically instantaneous or in other ways negligible in size, is with a finer word called differentiation.3 And the use of $\mathrm ds$ and $\mathrm dt$ to represent such infinitesimals is called Leibniz notation.4

Note, that

• we can call both $\frac{\Delta s}{\Delta t}$ and $\frac{\mathrm ds}{\mathrm dt}$ an ‘average’ – $\frac{\mathrm ds}{\mathrm dt}$ is just an average over a brief moment – and
• we can also call both of them a ‘change’ (a ‘change over time’) – $\frac{\mathrm ds}{\mathrm dt}$ is just an instantaneous change.

While this notation clearly shows the involved terms, it is a bit, shall we say cumbersome. Let’s invent a simpler symbol for ‘derivative’, why not just an apostrophe $’$, usually called a prime:5

$$v=s’$$

That is so much easier. If needed, we can choose to add an extra little lowered label with the property we are deriving to, so it clearly is ‘the derivative of distance $s$ to time $t$ ’:

$$v=s_t’$$

References:

1. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
2. Encyclopædia Britannica’ (encyclopedia), www.britannica.com