On the definition of the derivative.

🌟Support the channel🌟

🌟my other channels🌟

🌟My Links🌟

🌟How I make Thumbnails🌟

🌟Suggest a problem🌟

[swarley]

Quantum derivative sounds like something they would make up in a sci-fi movie to sound fancy and futuristic

[MichaelRothwell]

I think that to use the quantum derivative you need to specify that a≠0.

There is also a fourth definition, often used in complex analysis and for functions on Rⁿ, and that is f'(a)=φ(a), where φ is a function, continuous at x=a, such that
f(x)=f(a)+(x-a)φ(x).

[Saheryk]

- Dad I got detained.
- Why?
- Teacher asked what's 2*7 and then what's 7*2.
- But it's the same shit.
- That's what I said.

[diarmuidkeane]

these collection of definitions would make a great t-shirt Michael!

[guitaristxcore]

looking forward to that Quantum Derivative video

[SiiKiiN]

Sometimes I feel like I have a Michael penn in my brain whenever I try to read math explaining to me.

[tiskbubbles]

You can also take the limit as h goes to 0 of [f(x + h)/f(x)]^(1/h) which is also another derivative-like object, and in fact is related to the ordinary derivative as e^[f'(x)/f(x)], aka e to the logarithmic derivative. There are infinitely many of these. I'm not sure where they are treated in mainstream math but I have seen Grossman, Grossman, and Katz have done some obscure work on general derivative-like objects in what they call "Non-Newtonian Calculus" and "meta-calculus".

[rajendramisir]

Interesting and insightful. For the limit of quantum derivative, the inputs approach each other, multiplicatively. For the limit of difference quotient, the inputs approach each other, additively. And, when h=1, the limit is a forward difference operator or discrete derivative. Interesting differences. Thanks for sharing. Interesting comments.

[tomkerruish]

How about a video on the Fréchet derivative?

[HoSza]

My book be like f(x)-f(a) = A(x-a)+o(x-a). This generalises better for more abstract spaces.

[billtruttschel]

When h=1, my textbook calls it the "net change."

[farfa]

Well then can they also approach exponentially? And factorially?

[DmitriStarostin]

Each kind of derivative is applicable to a specific class of physics problems. L. Euler seemed to have understood the derivative as variant #1 since x approaching zero does not really work with logarithm axiomatically while variant #1 does. I think L. Euler's general premise in astronomy problems was later related to 'variation of Lagrangian equals zero'. 'X approaching zero' never happens in astronomy since there is no 'zero' in the systems that are part of complex rotational movements. It could be redefined, of course, but we get definition #1 then. Quantum derivative, it seems to me, is based on the postulates of the quantum mechanics, on the idea of the superposition of the wave functions and on the premise that a wave-particle should be somewhere, it cannot just fizzle out in a collider. So to me, the quantum derivative and Richard Feinman's technique are ultimately based on trigonometric functions (e ** (i * pi), colliders are ultimately rings), but they may not apply to metrics governed by other equations like those in GTR, or cubic and power-4 equations (like electromagnetic oscillator). That quantum theory and GTR did not work together well for a long time is a well-known story.

[krabbediem]

*cross my fingers, close my eyes, and whisper 'please-please-please be an old shorts, pleeease...'*

[Bjowolf]

Does this also include the "half" (etc.) derivative?
( as potentially used (defined) in LaPlace & Fourier transforms )

[shruggzdastr-facedclown]

A derivative is the slope of the line tangent to a given point on a curve, and an integral is the area of the space beneath the curve, right -- or do I have it backwards?

[puikihung]

can we define other derivatives by using other binary operations a*b, where a and b are elements of a continous group / spaces with * operation?

[apollo]

Cool, I remember using the multiplicative variant when I first learned calculus without even knowing it was a defined thing... I would love to see a video about it!

[ninolatimer]

What is the discrete derivative used for?

[King-sdvg]

This is the first time I hear of such things 😭