And here’s a quick recap of a demonstration of the product rule (after Leibniz):

Which is, of course, the product rule.

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# Tag: calculus

## Product rule

## Integration by parts

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## Euler’s formula proof

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## Derivative of any number raised to the power of x

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And here’s a quick recap of a demonstration of the product rule (after Leibniz):

Which is, of course, the product rule.

Saw this referred to in A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity and it made me shudder – as I always seemed to struggle with it at ‘A’ level maths – so here, for my own benefit, is a quick proof/explanation of the method.

The general rule is:

And why is this so?

From the product rule we know:

So

And, integrate both sides and we have:

Of course, we still have to apply this sensibly to make our problem easier to integrate!

Reading An Introduction to Laplace Transforms and Fourier Series I reach the point where it is stated, rather axiomatically, that: .

This is a beautiful formula and has always suggested to me some sort of mystical inner mathematical harmony (yes, I am a materialist, but I cannot help it).

But these days I also want to see the proof, so here is one:

We know that complex numbers can be described in polar co-ordinates:

So too where and depend on .

Now (and applying the product rule)

So we equate the real and imaginary sides of both sides of this equality we have:

Then, recalling , we have

By inspection we can see that and , giving us:

and multiplying both sides by we have:

Reversing the signs:

But what of and ? Well, we have and .

So is constant with respect to while varies as .

If we set then – a wholly real number, so and . Thus and we can replace with throughout.

Hence: .

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I know this is a piece of elementary calculus but I just worked it out from (more or less) first principles (as I knew what the answer was but did not know why).

Let what is ?

where is some constant. Hence or

and so and so .

And , so

Applying the chain rule and so

Which is

Now I have written it all out I know there are redundant steps in there, but this is how I (re)discovered it…

Not sure, so maybe someone who knows can tell me.

Following on from the last blog, can we show ?

Assume a constant, exists such that , could give us this ?

**Edit**: Professor Rubin (see comments) tells me that, as I feared, what follows is not supportable:

Now, and this is the bit I have most doubts over, as and then and and assuming then we are left with .

Hence and therefore , the constant we are seeking.

But there are a lot of assumptions in there: anybody able to tell me how valid they are?

This is just an online note to myself about differential calculus. A level maths again…

Calculating

where

Using the chain rule:

(as )

hence .

So

And , Euler’s number, really is magical.

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