Derivatives

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This simple, elegant design illustrates a crucial and revolutionary discovery in mathematics – the derivative and its close cousin, the infinitesimal calculus. It is the subject of entire university courses and weighty books. We have one picture and the equivalent of a sheet of A4 paper. Don’t expect miracles – or even an explanation of which a real mathematician would approve!

Look at your print and, since it comes in a variety of colour combinations, use the first version above as the key to the colours described here. The black line is a stylized graph of the value of $\tfrac{1}{3}{{x}^{3}}$ . You’ll have to imagine the axis lines that you usually see on a graph, which meet in the middle, where three of the curves cross at the point where $x=0.$

The grey line represents ${{x}^{2}}$ , the brown line represents $2x$ , the yellow line represents the constant 2. And there’s a reason we’ve chosen those particular expressions which, hopefully, will become clear.

The key word to remember when you hear the word ‘derivative’ is ‘slope’. Mathematicians have a simple definition for slope; it is ‘rise over run’. If a line goes ‘up’ two (it doesn’t matter what the measurement unit is) while going ‘along’ one, then it has a slope of ${\scriptstyle{}^{2}\!\!\diagup\!\!{}_{1}\;}$ , in other words, 2.

Easy to see what that means for a straight line, but what on earth does it mean, for example, for the grey curve – surely its slope is constantly changing. Well yes and no – it depends how you measure the rise. If all you do is measure the change in the value of ${{x}^{2}}$ when the value of $x$ alters by 1, you might well conclude that the slope is all over the place. In the middle there’s a zone where moving left or right makes very little difference. But move out to the sides at a small movement results in the graph shooting up or down.

But what happens if, instead of measuring the change in ${{x}^{2}}$ in terms of a simple number, we measure it in terms of $x$ itself.

We know how ${{x}^{2}}$ works, so when we add one to $x$ , the resulting value goes up from ${{x}^{2}}$ to ${{\left( x+1 \right)}^{2}}$ and that (you either know or will have to take my word for it) equals ${{x}^{2}}+2x+1$ . So whenever we increase $x$ by 1, the value of ${{\left( x+1 \right)}^{2}}$ increases by $2x+1$ .

That’s pretty neat but it’s not quite what we were looking for. It’s not really the slope of ${{x}^{2}}$ at any particular point, it’s the slope of the straight line from the point ${{x}^{2}}$ to the point ${{\left( x+1 \right)}^{2}}$ on the curve.

Now comes the magic. Suppose that for our ‘run’, instead of measuring from $x$ to ${{\left( x+1 \right)}^{2}}$ we measure from $x$ to the $x$ + ‘‘the tiniest fraction we can imagine” (call it $tiny$ ). What happens to the value of ${{x}^{2}}$ when we move from $x$ to $x+tiny$ ? Well it’s simple multiplication, though again you may have to take our word for it:

${{(x+tiny)}^{2}}={{x}^{2}}+(2\times tiny\times x)+tin{{y}^{2}}$

When multiplying fractions it’s often helpful to replace the word “times” by “of” in your head. So $tin{{y}^{2}}$ means the tiniest fraction of the tiniest fraction, effectively nothing, so we can ignore it. So when we move from $x$ to $x+tiny$ (the run), the graph moves up by $2\times tiny\times x$ (the rise). Since slope is rise/run, we have

$\displaystyle \frac{2\times tiny\times x}{tiny}$

The $tiny$ at the top and the bottom cancel out and the slope is $2x$ . And it’s $2x$ anywhere along the curve representing ${{x}^{2}}$ .

It’s exactly the same technique that allows us to say with certainty that the slope of $\tfrac{1}{3}{{x}^{3}}$ (black) is always and everywhere ${{x}^{2}}$ (grey), that the slope of ${{x}^{2}}$ is always $2x$ (brown) and that the slope of $2x$ is just 2 (yellow).

The simple, yet incredibly powerful insight which led to using unimaginably small values in calculations, is down largely to the genius of both Gottfried Leibniz and Isaac Newton (he of the apple). It can be applied to expressions of almost every kind and it revolutionized mathematics, science and engineering because what we’ve described simply as slope, in the real world is also rate of change.

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Derivatives

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