What Inspired the Derivations in Calculus?

[symbolipoint]

How did who decide some of the steps to choose for developing the Chain Rule and for developing the derivative of the Sine function? I spent some time during the last few days reading and examing these derivations, and although now I can understand those, the steps taken do not seem very obvious; meaningful and correct, yes so they seem; but how to know to choose those particular steps, not at all obvious.

Did those who derived these rules use special proof techniques which are commonly taught in some transitional or upper division courses? Were those people simply clever geniuses?

 

[christianjb]

You're seeing how concepts are taught- not how they were first developed.

It took two millenia from the time of Euclid for Newton and Leibnitz to begin to formulate calculus. It then took a couple of centuries for mathematicians to develop and really understand calculus.

 

[arildno]

Did those who derived these rules use special proof techniques which are commonly taught in some transitional or upper division courses?

They went by intuition as often as by logic.

Were those people simply clever geniuses?

I would certainly call Newton, Leibniz, the Bernoullis and Euler for clever geniuses, yes.

 

[Werg22]

They went by intuition as often as by logic.

I would certainly call Newton, Leibniz, the Bernoullis and Euler for clever geniuses, yes.

Euler and Newton a little above the others

 

[Data]

I don't know. I think it's pretty presumptuous to make historical judgements like that, about things so far removed from our present situation. There are thousands of stories of discoveries found and lost, or of misappportioned credit.

What people are doing today is based on what tens of thousands of other people have done in the past, and what people do in the future will be based on what tens of thousands of people are working on now. We're fortunate to have such a rich history of brilliant people to draw on!

 

[robert Ihnot]

The derivative of the sin(x) is not difficult taking it straight from the definition. Limit h goes to zero, $$\frac{sin(x+h)-sin(x)}{h}$$

You have to know that the limit as h goes to zero of sin(h)/h is 1. This generally is explained at the beginning of calculus study. Also the limit of sin(x)cos(h) goes to sin(x). So we work though the expansion: $$\frac{sin(x)cos(h)+sin(h)cos(x)-sin(x)}{h}$$ It is not a difficult derivative. However, I don't know of anyone who is credited with finding it.

As far as the definition of the derivative, some of that comes from the study of tangents, and Fermat took an interest in that. Newton and Leibniz are given the credit for understanding the Calculus as a algorithmic process and a general mathematical tool.

As far as the chain rule, in some cases that could be guessed without using the chain rule, say F(x) =X^2, G(x) = sin(x). Then F(G(x)) = sin^2(x). But we can directly find the derivative of the sin^2(x) since sin^2(x+h)-sin^2(x) =[sin(x+h)-sin(x)][sin(x+h)+sin(x)], so dividing by h and taking the limit the result is 2sin(x)cos(x).

 

[Gib Z]

In the days of Euler and Newton, the tricks were from the minds of mathematical genius. It is especially these 2 that used clever and not always the most rigorous tricks in the book. Euler with his solution to the sum of the reciprocals of the squares, and Newton which his geometric argument proving Kepler's Laws. Most other mathematicians seemed to be all about grinding through the abstract but computable works, whilst these 2 showed that they actually understood their mathematics. There are still people like this recently, such as Paul Erdos.

So yes, the people who pioneered their fields- Strokes of mathematical genius.

 

[Data]

I don't know why you think that you know that it's "especially" those two who came up with unique solutions to problems. People have done it throughout history and are still doing it today. Not everyone discovers calculus, but difficult problems are solved in ingenious ways every day. Euler and Newton (not to mention Erdos!) certainly contributed more than their share, but so have thousands of others.

Rigor was a rare commodity in many contexts back then, as well.

 

[matt grime]

Most other mathematicians seemed to be all about grinding through the abstract but computable works, whilst these 2 [Euler and Newton] showed that they actually understood their mathematics.

Would you be able to justify that comment at all? I'd start with you naming a dozen contemporary mathematicians/natural scientists, then a list of their abstract but computable works... (you might want to think about the Scottish contribution to mathematics).

 

[Gib Z]

>.< I doubt I would be able to justify my ignorance, probably due to my lack of reading. Perhaps many other mathematicians also had creative moments But I thought Euler and Newton were prime examples of them. I would also like to point out that many mathematicians in history are forgotten as the may have contributed something very minor or nothing at all, but nevertheless studied mathematics. It were those who I was referring to when I made my comment.

Minor addition, but I would also like to mention I appreciate Data says "discovers" rather than "invents" calculus.

 

[Data]

In Newton's case "invents" might be more appropriate, actually: He worked the fundamentals of calculus out to solve particular physical problems.

Here are some comments by none other than Newton himself, though:

"To explain all nature is too difficult a task for anyone man or even for anyone age. 'Tis much better to do a little with certainty, & leave the rest for others that come after you, than to explain all things by conjecture without making sure of any thing."

"If I have seen further it is by standing on ye shoulders of Giants."

 

[Integral]

...
"If I have seen further it is by standing on ye shoulders of Giants."

Now if you read John Gribbins, author of In Search of Schrödinger's Cat it may well be that that comment is actually a dig at Huygens (IIRC) who, physically, was something less then a giant. Remember that while Newton was proposing a particle theory of light, Huygens was pushing a wave theory. This led to a bit a animosity between the 2 prominent scientists.

 

[arildno]

Not Huygens, Integral. It was aimed at Hooke.

Newton and Hooke had a hearty detestation of each other.