The other way to visualize derivatives | Chapter 12, Essence of calculus
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when they want to show functions as transformations of the number line, for ideas like fixed points, continuity and warped grids.
Recommended by 3 people across 4 different months, 2018 to 2023.
What it is
A 3Blue1Brown video from the Essence of calculus series that presents derivatives as the local stretching or squishing of the number line under a function, rather than as slopes of graphs. It applies this view to an infinite fraction puzzle, cobweb diagrams and the stability of fixed points, with Grant Sanderson's animations throughout.
What people said
Recommends Arthur · 2018-07 · Mathematics Stack Exchange · on “Intuition behind chain rule”
@Thor Since I made this answer, 3blue1brown has made a video showing this visualisation better than I ever could: [this video] It may not go into chain rule, but it does do derivatives, and with that as a help, I think visualising the chain rule the way I describe here isn't too difficult.
Recommends Arthur · 2019-09 · Mathematics Stack Exchange · on “Explanation Of Continuity”
This visualisation becomes quite busy. I personally prefer to think about a different visualisation. This 3blue1brown video [this video] shows more or less how I like to think about it. It wasn't until I started visualising functions this way (general functions in real analysis can't really be visualised any other way, but I wish I had learned it in my first calculus class a few semesters earlier) that $\varepsilon$-$\delta$ really "clicked" for me.
Recommends Riemann'sPointyNose · 2020-07 · Mathematics Stack Exchange · on “Interpretation and use of the logarithmic scale for high school students”
If I remember correctly, this video does a really good job at this sort of thing: [this video] . What you have shown above is just a different way of looking at what exactly a function does. For example, let's say we had $${f(x) = x + 1}$$ This just "moves the number line left 1 unit" - but it does not affect the distance between two numbers. The numbers $0$ and $1$ have a distance of $1$ between them, and ${f(0)}$ and ${f(1)}$ has a distance of ${f(1)-f(0)=2-1=1}$ between them also - the distances stay the same! Another example is $${f(x) = 2x}$$ In this case, the distances between numbers are affected. Take the same example of $0$ and $1$ - ${f(0) = 0, f(1) = 2}$ , and so ${f(0)}$ and ${f(1)}$
Recommends Sai Mehta · 2023-01 · Mathematics Stack Exchange · on “Calculate the exact value of an infinite sequence”
Like @mowzorn said, it is a good idea to let $$x=1+\frac{1}{1+\frac{1}{1+\ldots}}$$ so that $x=1+\frac{1}{x}$ . You will see that there are two values of $x$ for which this works - $\frac{1+\sqrt5}{2}$ and $\frac{1-\sqrt5}{2}$ . It seems strange that this expression could have two different values. You can argue that only $\frac{1+\sqrt5}{2}$ is a valid solution but you could also argue that $\frac{1-\sqrt5}{2}$ is a solution to your problem. 3Blue1Brown made a video on this, using differentiation to show the two perspectives - the link is here: The other way to visualize derivatives | Chapter 12, Essence of calculus [this video] .
Cites Nya Legoo · 2021-05 · Mathematics Stack Exchange · on “How to find an equation that warps a grid system or transforms a number line.”
I am a Microbiologist with no Math background. I am looking for an equation or how to find equations that warp a normal XY plane in the Jacobian fashion. Like this [this video] I want to end up with an f(x,y) that returns f(x',y'). In other words, I want an f(x,y) that will allow me to see how a point (x,y) moves to (x',y'). I want, in my hand, a function that when I plug in (x,y) it spits out x',y'. Idea/Method. A Jacobian matrix tracks or should track the movements of all points. If I can get the partial derivatives, then I can integrate to obtain the initial function. In 1D, I'd like to think of a y=f(x) that returns a curve. So when I have the curve, I'd like to find f(x). I like to think of
Cites Nya Legoo · 2021-05 · Mathematics Stack Exchange · on “How to find an equation that warps a grid system or transforms a number line.”
@kimchilover, I have the book. I'm taking a look at it. I feel that it explains how a few functions work. I feel It's a little deep waters for me at the moment. But I'm trying. Regarding how my question is structured, we agreed with your analogy. Your face -> outline of Portugal. I think this is the perfect analogy of what I'm asking. [this video] You can see the normal straight grid and a warped one.