A New Definition of the Derivative?  

Just leaving a comment to address some common objections. Yes, this is the limiting case of the quantum derivative (mentioned at the end). No, it does not extend well to higher dimensions. Yes, the multiplicative definition is undefined at zero. There is a fix for this mentioned at the end, f'(0) = lim x--> 0 of f'(x), if this limit exists. But an an important caveat that I did overlook, is that even this 'patch' only works for functions that are continuously differentiable on their domain - an exception would be something like f(x) = x^2 sin(1/x). More importantly, I'm not advocating for this to replace the standard definition, nor claiming that is equivalent to the standard definition in all possible cases. I am presenting it simply as a novel way to think about the derivative, one that offers deep insight into the forms of the derivatives of certain functions - namely power functions and log functions, both of which are continuously differentiable. First, the algebra is much cleaner for these functions as the x's 'pop out' of the limits. As a result, the power reduction in the power rule is immediately obvious for ANY real non-zero power, and the rule itself can be easily proved for any rational power. In contrast, the standard proof (expanding the binomial) really only works well for positive integers. And for logs, the form of the derivative is seen directly: for any fixed value of t, the 'rise' is constant for all x, and the 'run' is proportional to x, so the derivative of any log function must be proportional to 1/x. We also get a nice closed-form derivation of |x| (which of course is not differentiable at 0) from first principles to boot. These are all concepts that could easily be shown to a bright Calc 1 student or class ALONGSIDE the traditional definition, with the appropriate caveats. But perhaps it's only useful to people who watch videos like this and enjoy revisiting math they've learned to understand it on a deeper level. Maybe I could have framed it differently and spelled out that I was more interested in understanding structure than going for mathematical rigor, as it is a youtube explainer, not a published paper. But that's really all this video is about - changing perspective to gain deeper insight into fundamental concepts.

as a rule of thumb, if your function plays nicely with addition, use the original, and if it plays nicely with multiplication, use this new one

As someone who thinks about pedagogy, I have a few thoughts! As you allude to in the video, this "new" definition of the derivative is really the same definition just with a change of variable. Getting this idea across is actually the crux of problem when it comes to teaching I think. At this stage in their academic careers, many students (in my estimation) still conceptualize substitution and change of variable in terms of a mechanism/algorithm rather than as a fundamental structure-preserving transformation, and this can block the clarity in the intuition that the two formulations are really equivalent, which can make the overall definition harder to grasp.

I think the most interesting thing to take from this video is that we can use suitable substitutions on the limit definition to get "new definitions" that can be easier for certain problems. It's a useful thing to have on the toolbox, specially for students just starting out. I know I didn't appreciate substitutions enough when entering uni.

This is EXACTLY the video I needed to understand the derivative!! I have never understood it before… but now I realize how simple it is!

My high school calculus taught us using that secant-> tangent method. Is seriously attribute all of my ability to intuitively understand calculus to that man.

You can actually extend this idea and use a custom way of approaching x based on the function you need to differentiate. Maybe using (sqrt(x)+h)^2 or e^(ln(x)+h) or some other expression can magically simplify the numerator without being too bad in the denominator

Speaking as a professional Calculus tutor, I think that, while some of the calculations are more intuitive, the definition itself is less so. I think the definition is in fact more important. After learning the definition, the student learns the basic functions and the rules for combining them. I think that demonstrating the chain rule through such a definition would be a real brain buster for students.

I love your video lesson about the derivative. Very neat, clean , very didactic and elegant explanations. I will keep both ways to find the derivative of a function nevertheless interesting to mention that in certain optimization problems We let the computer do the work [for example optimizing an objective function, performance function, cost function or solving and optimal control optimization...]. I will keep visiting your channel for more exciting mathematical definitions, strategies, etc...

17:00 is amazing. Thank you sooo much. It has been so many years. As a game developer, derivatives matter every now and then in my daily work. 20:00 blew my mind.

Its the same with hyperreal numbers. If you use them in the standard derivative method (as "h"), you do not even need any limit to get your result. All the "dx" magic (multiplying, dividing, etc by "dx") is often times viewed as bad practise. With hyperreal numbers, you can do "anything" with "dx" without invalidating any equation.

My teacher used to tell us that the right definition for derivation is f(x+y) = f(x) +df(x)(y) ||x-y|| + o(||x-y||) That definition using a notation similar to taylor series always made the most sense to me as it outlines what a derivative really is, it is just a linear approximation of a function. Hence that makes the binomial solution make a lot of sense because you only want to keep first order terms

A) I really like this idea for teaching the derivative. I am going to school for mathematics education and I truly believe that this would be more understandable for most students. B) You have a very calming voice! I might use this video to help me sleep in the future 😅

Very nice! Also explaining the limitations of your findings make them even more useful, I’m linking them to a friend of mine who teaches physics in high school, hope it’s ok

I like it. Caveats aside; Its another useful tool, and it has one valuable lesson; sometimes drifting ever so slightly from the railroad tracks we tend to be taught down can open possibilities of shortcuts in certain situations or a better understanding by just looking at the same thing from a different angle. Well done.

In quantum calculus, the quantum derivative is actually very helpful in certain circumstances

1- you will have problems in the examples with the new definition at x=0. 2- at x eq0 you will see that the new and old definition are the same for sufficiently good functions by applying L’Hopital to the new definition

As for the closed form of the deriviative of |x|, I found that by chance when I was first learning calculus. Basically, I noticed that logs often have an absolute value around their input so they can be defined for negative values, but when you take the derivative you find it's still the same because the derivative of ln(x) is equal to the derivative of ln(-x) which is 1/x. That means means the derivative of ln|x| is also 1/x which you can write dln|x| = dx / x. But using the chain rule you can also find dln|x| = d|x| / |x|. Which means we have d|x| / |x| = dx / x or d|x| / dx = |x| / x.

... Good day to you, At time 13.22 Instead of applying u = sqrt(t), we could also consider the denominator (t - 1) as a difference of 2 squares: t - 1 = (sqrt(t) - 1)(sqrt(t) + 1) ... and then cancelling the common factor (sqrt(t) - 1) of top and bottom, resulting in the expression of 1/(sqrt(t) + 1), finally lim(t --> 1)(1/(sqrt(t) + 1) = 1/2; resulting in the same outcome of 1/2sqrt(x). Very nice presentation and thank you for your educational math efforts ... Jan-W

For f(x)=aˣ, the x+h definition gives f'(x)=f'(0)aˣ, which I always found really neat! Never could I have expected similar elegant results for f(x)=xⁿ, f(x)=logₐ(x), and f(x)=|x|. As an IMO gold medal winner (2022) who regularly scrolls through RU-vid, I have seen quite a lot of math, and I don't think of myself as easily impressed. But sometimes, I see an argument like this, and I am completely blown away by its elegance and simplicity.

This is coming from setting h to hx, in the original definition, that is, a relative increment, not an absolute increment, then renaming 1+h=t: (f(x+hx)-f(x))/(hx) = (f(tx)-f(x))/(tx-x). The relative increment is used in Financial Mathematics, where x is the price of a stock, and the derivative is called the (unit-less) delta sensitivity of a financial product (e.g., options) price (represented by function f) that depends on x.

this is a lovely new definition, arising from a simple shift in point of view: rather than a decreasingly determined "step" [delta], you collapse the "proportional distance" from x to the second point nice!

This definition is equivalent to the other (if the limit exists in both case), indeed every way of approching x can be either done by x+h or tx. Since the real topolgy is second countable, in particular every point have a countable open basis, we may choose countable sequence to fully caracterize limit (instead of nets) now you can show every sequence approching x can be done by some t_i that you mutliply with x or by some h_i that you add to x, taking i--> to infinity you get x. Supposing limit of (f(tx)-f(x))/(tx-x) exist then the classic derivative exists and equals the same, you can have problem where the denominator is undefined but you have to change. That's is why I think the first definition of derivative is the best way, in other word the linear approach is better.

actually, t^4-1 has the factors (t+1), (t-1) and (t^2+1), which can be further factored into (t^2+i) and (t^2-i) on the complex plane.

Quite cool! I'll remember that trick for the derivative of |x|! I would say that a really key insight not discussed here is that of units. Units are not just for physics and engineering, even though it's a pretty inescapable analogy. I'd say that the key here (or the key of the q-derivative) is that you write f'(x) as (unitful stuff)*(unitless stuff). If x were measured in meters, we don't want our complicated math to have units in it, so we don't really want to write "x+h" because that forces h to have units of meters as well! Instead, we can formulate our problem as we vary the unitless parameter t. Then, we write our answer as (unitful stuff)*(unitless function of unitless ratios). I'm bad at phrasing this stuff without using physics language, but it's the same trick you'd use if you're solving the nonlinear ODE L*f''(t)=-g*sin(f(t)) (a pendulum). Even though the problem is nonlinear, it's easy to prove that the period of the pendulum is T=sqrt(L/g)*r(f(0)), where r is a unitless function of the initial angle. If we really really insist that I can't use the word "units" or "meters" (say I'm doing a graduate math course homework problem!), then I can simply rephrase things in terms of rescaling the variables I work with, like choosing to work with t*x instead of h! It also comes in handy when debugging numerical analysis code. But OK, I've ranted enough for one youtube comment, thanks for the stimulating video!!! :)

That derivation of the derivative of the natural log was remarkable. As a mere student, congratulations!!

Maybe the derivative isn't usually defined in this way because this doesn't generalize to higher dimensions all that well? We usually require the convergence along all paths, not just "linear" ones (tx only hits multiples of x)

Awesome video! I love the idea of teaching this alongside the classic definition.

I have never seen this form of the derivative. Thank you for sharing.

The beggining for another maths related channel? New sub!

I think the warning about limited options should have been put at the beginning, not the end. The "advertisement" at the beginning creates the impression of "super power", but at the end with the "fine print" we realize that the "product" is not perfect. This is math, science - no need marketing and "9.99" magic required. Great video! Great broadening of horizons! Good pedagogical approach and illustrations.

Interesting. Takes me back a few years/decades. Knowing that the aim is to provide an "instantaneous" value I understand the concept of taking a derivative. These days I just use the standard rules but in an idle moment I'll give this approach a try.

Yeah I had come across this quite early on actually, I don't have a formal education in calculus but from the stuff I do know there have been multiple definitions of the derivative I have come across, including the symmetric derivative which is lim h->0 (f(x+h)-f(x-h))/(2h), I found that particular one interesting because it actually gives a value to some discontinuous functions at their discontinuity that make a lot of sense.

Thanks for such vivid explanation.

This is great content. I hope to see more from you. Subscribed.

This just goes to show that school was never meant to teach you. If this definition was introduced to my calculus class, so many more students would have understood

You blew up my mind, and that thing made me comment.

It's a really nice concept. I do have a comment on the graphics: when writing functions it's better to keep the notation "upwards" instead of italics - it makes it easier to separate variables from function names. In LaTeX specifically (which is what manim uses under the hood), most common functions have macros, e.g. \sin, \cos, \log, \ln, etc. If there isn't, one can always use \text{function}. For exponentials in the natural base, one can write e^{x}, but that again makes an italic e. So instead, I personally define a new command: ewcommand{\eu}{\mathrm{e}}. Then \eu{x} is typeset correctly.

9:40 Regarding a proof of the identity x^n-1=(x-1)(x^(n-1)+...+1), you can also just note that (x^(n-1)+...+1) is a geometric sum and equals (x^n-1)/(x-1) which gives you the result immediately.

Differentiation made me sense to me when I realised it was a higher order function. For f, it gives you d(f, x) = my+c, that is the line/plane/surface that touches f at x when y=x. If you always set y=x, and you inline d, you recover high-school calc.

really love that presentation style thank you sir

Zero is a tricky problem where there are infinitely many `1/0` values, but only the single zero covering all the subtly different uses of 'zero'. Given maths typically starts with the idea of 'next' (or increment), then yes, the 'h' method is nearest to that. The 't' method works real well for regular folks who haven't yet reach the required level of pedanticism ;-)

This was very enjoyable. Disclaimer: I am neither a mathematician nor a teacher, so my opinion comes with no authority whatsoever. Convergence is hard to teach/learn correctly, especially if a student will be exposed to higher mathematics further in their education. As convergence is defined in terms of distance rather than a ratio, I feel that the traditional approach might be better at building consistent intuition. I'd probably still throw your method in here and there as an illustration of how things can be approached in different ways. It is always nice to see that alternative approaches give consistent results. Also, well done on the manim scripting, the video flowed very smoothly.

For a new maths channel, you've certainly emerged swinging.

sin(tx)-sin(x) =2sin((t-1)x/2)cos((t+1)x/2). We can factor a cos(x) out of the limit but this still requires the limit of sin(θ)/θ. Similarly, for the exponential, we can factor e^x and we’re left with a numerator of e^((t-1)x)-1 which still basically requires the same method as the usual proof.

Excellent video! Very refreshing!

An interesting aspect regarding the topic is that you can define the derivative in Banach spaces using the Frechet derivative, which corresponds to a continuos linear functional between the spaces you are working with. In particular, if you have a function that maps from a Hilbert space to R, that function is an element of its dual space. By the Riesz representation theorem, there exists an element in the space that represents the function. If you consider the norm of the Hilbert space as N(x), then for DN(x) h = with < , > the inner product, the representing element is DN(x) = x / N(x). Taking the Hilbert space to be R with the norm N(x) = |x|, it recovers that DN(x) = | - |' (x) = x / |x|.

Love the video. I enjoyed it very much. Just one minor nuisance. While watching the video for the first time in a stop-motion fashion (i.e., stopping after every point and studying the formulas) I got confused at 20:13 where you convert 1/u * ln(1+u) to ln(1+u)^(1/u) with an animation that seems to indicate [ln(1+u)]^(1/u) since the first expression is log of (1+u) and the animation just moves the 1/u part right above the entire expression with out any other hints. After a while I figured out that this is not what you intended but it would have been easier to follow if you could add some extra brackets, as in ln[(1+u)^(1/u)] or some other indicator. Again, thanks for a good video.

about a year ago i had to prove the derivative of √x and it took me hours, so when it showed up in the video, i was overwhelmed with emotions i hadn't felt in a while. university level maths is no joke, and i'm glad i'm just studying computer science with a little bit of math on the side.

A different perspective. İmpressive 20:59 After that... OK REALLY IMPRESSIVE

Michael Penn talked about this so called "Quantum dereivative".

Calculus was my fav back in university years oh my when we got introduced to the L’hopitals rule

I was taught a different way to calculate differentiate functions using h terms and you take O(.) of the terms to find the derivative. I don't know what this is called but I can find a reference if anyone is interested.

Yooo, this is super dope! I've never really thought about redefining the limit-definition of the derivative to make it lend itself better algebraically to certain functions. Absolutely eye-opening and interesting! Now I'm wondering about perhaps other ways of extending the definition... (E.g. instead of an additive or a multiplicative change in `x`, perhaps one considers a different change in x, such that the algebra works out. In general, you'd have `z = g(x, h)`, where `g(x,0)=x`, and `g` takes whatever form is convenient for what you're trying to differentiate).

As an Engineer who has been exposed to a lot of math in my life, I find this video fascinating and I love this new approach. That said, I don't think that this is how I would have liked to learn derivatives the first time I came across the concept. Delta(y)/Delta(x) which converts to (y1-y)/(x1-x) and from there to [f(x1)-f(x)]/(x1-x) defines the concept. And from there to call x1 = x+delta(x) there is only one step. But it would not even be my immediate step, I would stick with the "moving" point approaching the reference point for a while before moving to the delta approaching zero. For example, for y=x^2, dy/dx = lim |x1->x| (x1^2-x^2)/(x1-x), working with the difference of squares we get lim |x1->x| (x1+x)(x1-x)/(x1-x) = lim |x1->x| (x1+x) and now we can replace x1 with x giving (x+x) = 2x.

Good video. Nice way to advance new ideas. Any thoughts on the derivative as nothing more than the slice of a higher dim object at a given point? pack them all back together (integration) to get the original strucutre?

Very well-put together! Personally I'm not entirely convinced of its use---to me, a lot of the nontrivial examples end up needing tricks or arguments that are roughly equivalent in trickiness/difficulty as in the original method's proofs anyway :)) So I don't really see any advantages---through intuition or ease of rigor---that this alternative definition provides. But I can still appreciate a video with clear explanations and great visuals, and this video is certainly excellent! The video was paced quite well and you have a voice well-suited for voiceovers. Some particular comments I have: -Re: |x|. Gonna start with a pet peeve of mine---people not liking piecewise functions and somehow thinking "closed forms" are superior. I would prefer to disabuse people of this notion as soon as possible, actually!!! Is there any particular reason for |x|/x to be preferable over the +/-1 piecewise? I'd argue no, and In fact, I'd argue it's worse, because it's just needlessly obfuscating the simple actual intuitive thing going on (-1 slope when negative, +1 slope when positive). Justice for if statements!! Elementary closed forms are tooooootally overrated, and I don't want to encourage my students to put them on even more of a pedestal :)) Also, computing the derivative of |x| using the standard definition is not at all hard. The only technical bit is needing to argue that when x!=0, we know that |x+t| and |x| have the same sign when t is sufficiently small----an argument that your alternate solution also still basically had to make, by the way, when you argued that |t| = t as t -> 1 :PP -Re: sqrt(x). Again, your argument is of equal complexity to just using the standard definition of the derivative anyway. You still had to use the "conjugate trick" in the end to ultimately resolve your limit expression (while the substitution trick is clever, that's... just doing the conjugate trick, but wearing a fake mustache). - Re: Power rule. I really don't see what's so unintuitive with the "binomial expansion". In fact, the argument here is the beating heart of understanding calculus intuitively: recognizing that (x+h)^n = x^n + nx^(n-1) h + (a bunch of other terms that aren't important because they go to 0). In fact, it feels disingenuously "over-spooky" to say that it _needs_ the binomial expansion and to write out all the binomial coefficients (you even expanded C(n, 2) into n(n-1)/2, what is the point of that other than to scare people??), when most of the coefficients are actually completely irrelevant and don't need to be written out. (What follows is a somewhat weird take I have which is off-topic from general Calc 1 pedagogy) Also: you can prove the power rule for rational exponents i.e. x^(a/b) by proving x^a and x^(1/b) separately and then invoking the chain rule. And then it's not immediately obvious how to extend this to real numbers... which I like! Because it forces students to think critically about what the heck something like 2^pi is _even supposed to mean_!!! I hate how much weirdness about real numbers just gets swept under the rug in math classes, since many students (my past self included) end up with the feeling that real numbers are totally normal things, when actually they are Incredibly Weird and that rigorously defining things with them is Incredibly Tricky And yes, I'm talking about rigor here. Because if we disregard rigor, I'm pretty sure most people would already be convinced that the power rule holds in general, because it holds for integers and rational exponents, and like---polynomials are nice, so i wouldn't expect them to be too weird and suddenly break. So the only time (imho) we're ever actually discussing details is when in search of rigorous definitions and proofs. My understanding of the typical proof of power rule for real-number exponents is to let y = x^t imply ln(y) = t * ln(x) and then do implicit differentiation---which isn't so tough at all! Which finally leads me to... - Re: ln(x). In many calculus classes, ln and exp are held off until the integral lesson, and then it is _defined_ to be the antiderivative of 1/x and e^x is defined as its inverse (from which many familiar properties of it and exp(x) can be derived). Once again, I emphasize that you somewhat _need_ to do this because otherwise there is this gigantic gaping hole of what a real-number exponent _even means at all_. Your proposed solution here takes a shortcut by just taking for granted that ln(x) and exp(x) exist and are things we can use... when they really haven't been defined yet at this point in time, and to be frank "I have no idea what they mean" :P My preferred intuition is to define exp(x) by its taylor series, but obviously that can't be used until Calc 2. I suppose you could go define e.g. 2^pi as the limit of the sequence 2^3, 2^3.1, 2^3.14, 2^3.141, 2^3.1415... but that involves a lot of handwaving about continuity that makes the argument feel weaker and more wishy-washy (freshman me sure wouldn't have been ready to deal with Cauchy sequences back then!!!)

Feeding the algorithm 🥘! I just learned about the q derivative from another channel, so this was a cool follow up.

Blew my mind! I'll be retaking calc 2 & 3 this year, so this will be really useful.

I came up with similar unorthodox methods of computing derivatives using the limit definition and was I was rather scoffed off by my peers and shunned by the teachers, I expected them to be a bit more supportive and guide me thru the technicalities and making the process more rigourous. But oh well we dont have time for that when we're preparing for competitive exams🙅‍♀️

You can actually work out a closed form for the abs value of x using the original definition. Just sub in |x| = √(x²) and you can do it.

This is a really clever approach. Thanks for sharing it.

I like both definitions, and I'd use them both.

I can see this working like a math experiment in a classroom-setting. First, you demonstrate that the secant approaches the tangent (I think Newton’s lemmas in his Principia work well for this step). And then you ask how that might play itself out mathematically, and let your students play around with different models. This multiplication definition might arise, even though I would the majority of solvers to figure out the addition version first.

The reason it's not taught is really simple. Mathematician love generality, but the new definition only works for Real functions. In multivariable calculus, all paths have to have the same limit point, but with f(tx)-f(x) there is only one path possible: along the direction x points to (the existence of partial or directional derivatives does not guarantee a total derivative). In multi.dim. it has to be possible to independently choose a point and a direction and now it's the typical derivative or a directional derivative in the 1D case, since the only positive oriented normed basis vector is 1 itself.

I suppose this would be a good way to re-introduce limits in multivariable calc when considering limits along different paths. As far as intro Calc, doubtful it will really make things any simpler or easier to grasp.

Beautiful and intuitive 👏🏾👏🏾👏🏾

Pretty nice video! Minor reflection. It seems like for the examples you essentially get to replace binomial expansion and factoring an h with doing polynomial division. You kind of hand waved the polynomial division. But that doesn’t actually seem simpler. It’s not exactly harder either. But I see little gain here. Also, this totally breaks when x=0. Finally, you might argue that the more difficult bit about derivatives is the concept of a limit and rules regarding it. That’s because it’s usually not very clearly defined at this level. The sqrt, abs and log results were pretty nice. Always nice to see multiple perspectives.

amazing video!! definitely deserves more views

great vid, i look forward to your next upload

It's quite nice that t is unitless.

Amazing video!, so comprehensive

Essentially if ii understood well, what just happened is that the 't' factor defines the form of the coefficient to the X^n-1 term while the x term(exponent carrier) defines the form of the main derivative. we could even render that more rigourous by studying all such forms gotten from the algebraic mouvements expressed by factor 't' if we define some set "C" for (coefficient ) that can be adjoint to some set "F" for (Forms); all such forms carried by the main exponent carrier x. This could help us meausure more rigourously advantages in speed and other factors such as cleaniness of steps that arise due to the two methods. Donot know if all what i said makes sense but if not, we here to talk and contribute. Those were my thoughts.

Very nice video. However, correct me if I'm wrong, this approach would not be easily extended to the multidimensional case.

Very cool! As someone that gets the standard form conceptually, while also being rubbed entirely the wrong way by it, having this other option makes me feel much less anxiety at the prospect of doing them but hand... Would be cool to see a similar alternative for integration or some other related topics

Beautiful approach. Breaking down the process into a simpler method is what makes math so exciting.

Some comments: * When deriving the derivative of |x|, I think you did use the piecewise definition when you said as t->1, |t| is t. * For the ln(x) derivative, you went from needing the chain rule to needing to know when to interchange logs and limits as well as knowing how to expand the series (1+1/n)^n.

this is real clever. lovely!

I've derived a similar closed form version of the derivative of absolute value of x by using abs(x) = sqrt(x^2) which is the vector way of writing it lol. Interestingly, you get the fraction the other way around, as x/|x| instead.

I like your new definition. Well, I like all math. I was exited to see how the h->1 defination would work for e^x. In the end, still a nice video. Good graphics and examples.

Amazing…looks this may open up a new horizon?

this definition seems great for functions which contain multiplication

Here is the case: If the functon f is multiplicative i.e. for all x, y we have f(xy) = f(x)f(y), then the derivative is f'(x) = f(x)/x lim [f(t)-1]/(t-1) where the limit is just a number which does not depend on x. This is a nice property but I think there are quite less functions which preserve the multiplication structure of R. Maybe they are precsely the functions of the form xˆa. Any counter examples?

Nice I Learned Again Derivatives

Wow fabulosa definición para introducir los temas

A reason why I think this definition is used less is that it doesn't generalize well to multiple dimensions. As soon as you have multiple dimensions your limit needs to be able to approach for every direction. In 1d this is fine since q can fluctuate as q1. In multiple dimensions you would need to define some multiplication on vectors which can be quite tricky.

Thank you! I finally understand how calculating a derivative works! It's real nice how you show multiple examples with both the conventionally standard method and your alternative! Watching this is like practising along almost.

Thanks for spreading this method bruh. 🤪👌

Really elagant, I like it!

I was hoping for you to mention that this method is like saying f'(x) = g(x).f'(1) Where g(x) is generally easier to find in shown cases, and f'(1) is some limit we'd have to solve 😅

What was the difference between the first and third(alternative) definition? They look literally the same to me?

As a challenge problem - try using this definition to prove the power rule for rational exponents: f(x) = x^(m/n). It's something the standard definition doesn't do easily.

I think if you use sum (difference) to product formulas, you can get the derivatives of sine and cosine. But you would still need to know sin(u)/u -> 1 as u ->0.

This was really cool to see, really helped me better understand the original definition of derivative to. Do you know if these definitions are equivalent, or a more formal math proof of this new derivative? I know the x =0 difference might make this a limiting case. But if it is an equivalent definition for functions differentiable at x=0, it would be really useful to teach as well.

You went from an additive delta variable to a multiplicative one. I wonder if you could continue this to other operators, like limit as h approaches 1 of (f(x^h) - f(x)) / (x^h - x). Are there certain functions for which such a definition is actually easier to work with?

While I was thinking about "where does this q derivative come from?!"....

This is the limit as q goes to 1 of the q-derivative, yes? Edit 2: oh, you mentioned it at the end Edit: Hmm, can something like this definition be made to work in an algebraic-geometry kind of setting, without a limit? I know there’s a like, way of defining linear approximations of a curve at a point, by like... taking the degree 1 component of a polynomial when expressed as based at the point at which one wants to take the linear approximation, uh, Hm. Idk how I was thinking to make the connection. I guess I was just thinking, “if we are working in F(x) , then the whole, ((q x)^n - x^n)/((qx)-x), uh, Well, I guess if we have that (q-1) being invertible, then that expression makes sense... But then we want to choose that to choose q to be 1 and therefore make in not invertible, uh.. well, the idea is to have the (q-1) cancel out, so... I guess if we can have something like, “on the set q≠1, we have this being equal to this other thing, and we look at what that other thing is at q=1”, but what’s the like, careful algebraic geometry way of saying that? I’m not sure. There should be a restriction map from “functions defined on the x,q plane” to “functions defined on the x,q plane except at q=1” and uh, Hm, I guess maybe some kind of density argument? Idk

Great video!

Would you happen to know an equivalent alternative to integration?

This is Fermat's method of derivatives.