Thoughts on derivative video?

As a current 2nd year maths and physics student (apologies if I'm then incorrect with anything I say) there are a couple of things I found very important and helpful when being taught. Functions essentialy model what is happening. Through the Taylor series, when we add more and more derivative terms our prediction becomes more accurate. When we take into account the curvature (2nd derivative etc) How the derivative related to geometry and dimensions. This can then be linked to linear algebra and determinants (e.g. 2nd order derivative test) Linking the two helped me (think I) understand electromagnetism. These should more than likely be covered in later videos but I think for conceptual visitation they are too important no to mention Love your work

If you want to explain the formula (uv)'=u'v+uv' I suggest the following: First, imagine u(x) as a horizontal segment with length u(x), and v(x) as a vertical segment of length v(x). You get a rectangle. You are in the control room, where you change the value of x by a small ∆x and you observe what happens to the area of the rectangle. Say u(x) and v(x) increase a little bit. Now the meaning of u' is: How many ∆x-s can we fit in the small increase of the horizontal segment? (Say we can fit 2.35 ∆x-s). On the other hand, the meaning of (uv)' is: By how many ∆x-s did the area of the rectangle increase? Now the formula follows directly from the picture! The two small rectangles have areas of vu' times ∆x, and uv' times ∆x, respectively. The very very small rectangle has an area of v'u'∆x times ∆x, so it contains a really small number of ∆x-s and we ignore it. Overall, the area of the rectangle increased by u'v+uv' ∆x-s, which is the desired result. You can do a similar animation that explains (1/u)'=-1/u^2. Here the equation comes from the two small rectangles having equal areas. From these two, you can now derive (u/v)'.

I'd say one of the non-standard intuitions I would want to see on your videos is the idea of differentation as approximating the function with a linear function. That sounds obvious in 1D calculus, but it makes multivariable calculus a much more intuitive continuation into higher dimensions

Zairaner

It may be a trivial point, but one of the best "aha!' moments for a student of mine was when we took a bird's eye view of what f'(x) = \lim_{h \to 0} \frac{ f( x + h) - f(x) }{h} actually meant. Obviously it's just a ratio of infinitely small changes in x, but it helped this student see how mathematical equations could be constructed to illustrate a geometric idea. This was for a high school student and is anecdotal, but maybe it is useful. Thanks!

I really enjoyed the geometric approach taken in the introduction series. Perhaps showing derivates with other shapes and then explaining them as well (for example, the derivative of x^2 = 2x as a square adding on 2 x-lengh lines for each dx you increase by, or showing that the derivative of a*x = a by showing a rectangle adding on lines of length a for every dx you add on) (of course I'm sure I'm too late at this point, but whatever)

You might want to have a look at the slideshow animations in this article, in particular the one that starts with the red car. <a href="http://acko.net/blog/to-infinity-and-beyond/" rel="nofollow noopener" target="_blank">http://acko.net/blog/to-infinity-and-beyond/</a>

Max Goldstein

I always wondered what the point of calculus was. Would be cool if you can explain how calculus came about and why it is useful.

I think it could be a good idea to build up a little of the theory behind the differential either here or during integration. I didn't actually get what was going on there until I made it all the way to the differential vs the gradient.

Just tell what calculus is as it is :) I always found that the best explanation is not something that makes it looks easier then it is (it cannot!), but always, the detail of the steps taken to comprehend the thing. Do not make a hole in the logic, and if we have no other choice than to do so, for an obvious reason that it's too complex, tell it. Your video about topology is one of the best, it's amazing, and I think it's a good idea to follow that same kind of narration.

The go-to for a lot of people is looking at a position-vs-time graph, looking at the average velocity between two points, and taking smaller and smaller intervals about a central time period as a means of defining instantaneous velocity. While this is the classic "secant line converging to the tangent line" idea, the physical intuition (and the possibility for generalization) is powerful: the smaller the time interval we look at, the more and more uniform motion looks. It is especially intuitive in this context because an accelerating object simply doesn't have time to change its velocity if we look at a small enough interval. Of course, this extends well to linearization and much more... but my favorite part of this has to be the natural introduction to convergence of sequences.

I think the most intuitive way to understand derivatives is through the lens of driving a car. If you glance at the speedometer, the speed that it reads is the instantaneous velocity (1st derivative) evaluated at time t=now. The slope is the average velocity, which is a more crude approximation. My average speed could be 30 but I could swing wildly from 50 to 10 regularly. Hitting the gas pedal illustrates acceleration, etc.

If you haven’t read it, take a look at Lockhart’s book _Measurement_.

Jacob Rus

My biggest concern (since I have utmost faith in the animations) is notation. While visual intuition is good, there are plenty of people who benefit more from having a clear set of definitions. For instance, I think the derivative should be formally defined as the limit of the difference quotient. This way, people who haven't seen calculus before will have something concrete to use when trying to learn the rules of differentiation. The various rules of differentiation do not seem intuitive (to me at least). I mean, the power rule seems to defy intuition for e.g. x^3 --&gt; 3x^2. I just need to plug in the difference quotient definition to make sense of it. Also, the difference quotient generalizes nicely for defining the derivatives of functions between manifolds. Building on the ideas from the linear algebra series, you can define the derivative at point x as the unique linear transformation A satisfying lim ||h|| --&gt; 0 of || f(x + h) - f(x) - Ah || / || h || = 0 This can be visualized very nicely by taking an arbitrary vector at a point on some surface, then taking the difference of the function at the two endpoints, then trying to approximate the change as the vector gets infinitesimally small in magnitude (but it may approach from any direction!).

Duncan Fairbanks

But I digress. To answer the question: I agree, the best illustration of derivative I've ever seen (or used) is referencing the position, velocity, and acceleration of the car. For instance: why is the derivative undefined when a function isn't continuous? Because when you teleport, there's no such thing as speed. This lets you justify the Intermediate Value Theorem rather nicely too, with reference to aircraft-based speed traps or certain speed cameras.

Eric Astor

I have an (admittedly biased) view... but I think that calculus intuition can be better taught by directly using infinitesimals, with approximating tangent lines as a useful application. We DO have a rigorous number system including infinitesimals - it just needs a hierarchy of infinitesimals, so we don't accidentally introduce new solutions to x^2 = 0. It's called the hyperreals. Most presentations focus a bit too heavily on justifying their existence... but the rules are relatively natural. (When I teach calculus, I just refer to infinitesimals as "tiny numbers", with special intuitive rules for when combinations of numbers end up tiny.) This allows the development of limit (and derivative, and everything else) without actually requiring that picture Brando refers to. You still use it, but the DEFINITION of limit is independent.

Eric Astor

I think one of the issues as it was pointed out by someone else is the confusing thing that derivatives are always portrayed with continuous lines so continuity and differentiability are "the same". This seems to be a very confusing thing and the first time its shown to you that its false its sort of disastrous, its as if you've been doing calculus for 4 years with the **wrong** intuitions and ideas. This might be because the concept of a limit is never explained well without resorting to that picture. Not sure how to fix the issue but it would be awesome if at some point the intuition that this essence of calculus builds doesn't seem invalidated once a concept like that is introduced. Instead of fixing misconceptions it would be nice if the foundations are built correctly since the first day. Hope it helps :)

agree! It would crucial to fix this confusion. Since teachers always draw continuous lines to draw derivatives this confusion comes up. It might be because the intuition of a limit is never correctly explained without the need to draw continuous lines (Khan academy has this issue for the definition of a limit). Maybe the need is to have a new way to explain the concept of limit.

I agree, I think it would be crucial to explain this point. Its as if it was the side of the same coin but it was never clear why they where in the same coin in the first place.

For me, the most intuitive explanation was simply mapping between pressing the gas pedal to the position, velocity (first derivative), acceleration (second derivative) and jerk for the third derivative.

For me, the most intuitive explanation was simply mapping between pressing the gas pedal to the position, velocity (first derivative), acceleration (second derivative) and jerk for the third derivative.

"I mean, so what?" I agree! I believe it would be helpful if, in the video, Grant would show lots of examples for what the x-axis and y-axis represent. I was recently helping a student prepare for their SATs, and the student was blind-sided by the fact that, in a graph relating years (x) to # of manatees (y), the slope was the per-year rate of manatee population increase. In another example: test-subjects' such-and-such metacarpal bone length is related to subjects' height, and the student is asked to infer - again, from the slope of the best-fit curve - what increase in height should be expected from a 1-centimeter increase in such-and-such metacarpal bone length. Other examples: the classic "position versus time" and also, "production cost versus quantity produced". This last one is a great example because the derivative (slope) is the marginal cost, a relevant economic concept, and also one might glean some intuition into the FTC by noting that total production cost is the sum (integral) of the marginal production costs. In conclusion: student intuition might be increased by finding a great variety of applications in which the x-axis and y-axis mean very different things. Maybe. I haven't tested this hypothesis in the classroom.

Jacob Mirra

The intuition I like the most for derivatives is along the lines of 'we take a small step in the x direction and measure the corresponding step f(x) makes'. In one coordinate this directly leads to the standard 'slope of the tangent line' while in more dimensions it gives a solid intuition (it seems obvious if you think about it this way) for why the derivative has to be a (for example) 2x3 matrix if you go from a 3d space to a 2d space: you can take 3 different steps in x and each yields a step(tangent vector) in the 2d output space. (Also highlighting the fact that the differential is, in fact, a linear map) Another + of this is that, technically speaking, d/dx literally is exactly that, namely a tangent vector of the x-Axis, and df/dx is a tangent vector on the y-Axis, so the intuition can't really fail in more complex situations the way some other intuitions may when you don't work with functions R-&gt;R

Jan Nienhaus

Here is one approach for animating the connection between the graph of a function f and the graph of its derivative f'. Take the graph of f, and for a particular point (ex. x=3), one may wonder where is the corresponding point of f'. Step 1) Draw the tangent line at x=3 and intersect it with the x-axis at point A. Step 2) draw a point B, one unit to the right of A. Step 3) construct the right triangle ABC (C is on the tangent line, ABC is the right angle). Step 4, move the height of point C to x=3 and here you go - this point must lie on the graph of the derivative.

A "Eureka" moment for me was when I realized that x/0 had a meaning, being the slope of the vector (0, 5), and arcsin(x/0) is just Pi/2.

Mark LaJoie

It might completely off topic for this video, but to me, the most important - and counterintuitive - property of differentiability is that it has little to do with continuity. When one draws a curve, it is "always" continuous and "always" differentiable leading to the false belief that one can always get a derivative from a curve or a function. Showing a fractal curve as an example of "always continuous, never differentiable" curve (may be as a side video) could be useful.

I'm going to offer something different - perhaps a headache :P- because instead of sharing an intuition I would like to describe a lack of intuition on my part. You might want to address it if you think it's worth it! The usual intuitive idea of an integral is "area under a curve", and the basic intuitive idea behind derivatives is usually "slope of the tangent line of a curve in a point". I have no problem whatsoever with any of those concepts independently. However, those two concepts (derivative and integral) are supposed to be tied together - something that comes up frequently in physics, when you can "travel up and down" between space, velocity, acceleration, etc. by derivatives and integrals. They seem to be inverse operations, just like addition and subtraction, or multiplication and division. That's where my lack of intuition shows. For, say, addition and subtraction, it's clear how traveling forward in the number line and traveling backward in the number line are opposite concepts. But how is the area under a curve the "inverse" of the slope of a tangent line of a curve in a point? What idea ties those two separate intuitions together? what intuitive justification can there be for them to cancel each other? That's what I have trouble visualizing.

I think there are two linked concepts that are normally not well captured by students. One is the idea of a ratio, simple as it may seem, students tend to see something like speed =distance/time as a formula that gives the right answer. It is easy to solve correctly questions on proportionality but actually not have a deep understanding of the concept. The other thing is the concept of change. I am also a physics teacher and I find that students need many different examples of things changing. So, if the position of an object changes, the change in position divided by the change in time gives you the average speed and as the time interval gets smaller, that average speed approaches the instantaneous speed. I think the insight from physics helps a lot in calculus, not just with kinematics.

Javier Almeida

What really got the intuition to stick for me was to think of it with a physical analogy, like driving a car for example - when i understood the relation between your speed (or strictly speaking, velocity) and the way your position changes, I found i could abstract that notion to other things - thinking of the derivative as what you do to the "gas pedal" of the function. I've used this intuition when I've tutored people in Calculus, and it seems to do a good job in making this abstract slope of the tangent line thing stick a bit - "when you're not moving, what's the slope of your position vs. time? 0. and as you put your foot down on the pedal, what happens? your position changes positively, and the slope is positive"

V

I agree that "it's the slope of the tangent line" is a useful connection to make, but I'd want it to go deeper than that. I mean, so what? So it's the slope of the tangent line? Big deal, right? What I'd want is for the video to clearly articulate the connection between the derivative as the rate of change, and *why* it just so happens that the rate of change corresponds to the tangent line. I also remember being pretty fascinated in my own calc classes by how derivatives decrement the exponent and integrals raise the exponent. There was something mysterious to me about this relationship--though it makes perfect sense when viewing those two operations as inverses of each other. Perhaps this is a subject for a different video, but I'd really like to have a better conceptual understanding of why derivatives and integrals are so closely linked to the exponents in polynomials.

jason black

Yeah, that's a gold mine of intuitions

Myles Buckley

In my undergrad PDEs class (many years ago), I remember the professor saying, "And you all know what a derivative is..." In my head, I thought of the standard Calc 1 rule for the derivative of a polynomial. But then he said, "It's the slope of the tangent line." And he drew a picture. That geometric intuition was eye-opening, and it's stuck with me to this day. I suspect your video would have something similar, but I relay the anecdote anyway.

Not a calculus teacher, but I remember when I was taking my first calculus class the idea of an infinitesimal I just accepted and always kept coming back to what that even means as I went all the way through multivariable calculus. When we used the notation of a infinetismal change with respect to another infinitesimal change I just told myself that somehow a line could be made with two point infinitesimally close, definately not one point guys, and we get a perfect tangent line somehow. I understood when the points got closer and closer it got closer to the tangent line, but that felt more like a observation and less absolute and perfect. Later I found this resource which helped to build some intuitions on the subject: <a href="https://betterexplained.com/articles/calculus-building-intuition-for-the-derivative/" rel="nofollow noopener" target="_blank">https://betterexplained.com/articles/calculus-building-intuition-for-the-derivative/</a>

It may not be much, but after tutoring many students in calculus, here is what has worked best for me personally: I draw y=x^2 and explain that we want the tangent line at the point x=1. Then I explain that we need two points to make a line, so we can make secant lines that approximate the tangent line. I make one point at x=1 and several others on the right that get closer to one, and I show that the secant lines approach the tangent line. Then we can take advantage of a limit to make those two points approach each other and get an exact tangent line.