What's so special about Euler's number e? | Chapter 5, Essence of calculus
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it when a question about the limit (x^n-1)/n, the definition of ln, or the derivative of exponentials is really asking about the intuition shown in this video.
Recommended by 0 people across 0 different months, 2017 to 2024.
What it is
A 3Blue1Brown animated explainer from the Essence of calculus series on why exponential functions are proportional to their own derivatives. It builds up the idea of e, the natural logarithm, and why writing e^(ct) is a choice of notation rather than a necessity.
What people said
Cites Arthur · 2017-12 · Mathematics Stack Exchange · on “Why does $\lim\limits_{n\to 0}\frac{x^n-1}{n} = \ln(x)$?”
If you're watching the video that I think you're watching ( [this video] ), and if I recall correctly, then there is nothing to prove here. That limit is some number that depends only on $x$, and he chooses to call that number $\ln x$. Of course, you could then ask how to prove that $\ln$ defined this way is the inverse of the exponential map. That's a different question, and one that, in the light of the answers given here, should be asked more carefully and with greater attention to exactly what you're given and where you want to end up.
Cites Adel · 2021-01 · Mathematics Stack Exchange · on “A alternative geometric (visual) solution to the derivative of square-root”
In this [LINK] video, 3Blue1Brown shows how one can derive the derivative geometrically. Around 2 min 30 sec, he shows how the derivative of $f(x)=x^2$ is $f´(x)=2x$ by starting with a square. The key idea is that the change in $f(x)$ , denoted $df=dx^2$ , captures the change in area. This change has three components: the two rectangles and the tiny rectangle. One obtains the area of one rectangle by multiplying the two sides, that is $xdx$ . The area of the tiny area is $dxdx$ . Thus, the change is area is then $dx^2=2xdx+dxdx$ . As the derivative is defined as the ratio between change in the input (i.e., $x$ ) and the output (i.e., $f(x)$ ), we solve for $dx^2/dx$ . By algebraic manipulati
Cites Andrew Thorson · 2024-06 · Mathematics Stack Exchange · on “When Does ((n^a)-1)/a)) Equal e; A Sophomore's plight”
I am a high school student (sophomore) and have come across something I would like explained. I was watching 3blue1brown for an explanation of calculus, when he used the formula: lim a->0 (d/dx(n^x)=(n^x)((n^a)-1)/a)). (7:05 of [this video] ) He was putting in different numbers for n, lamenting in the fact that ((n^a)-1)/a) approached some constant, which was ln(n). However, this sparked a question of what value n has to be in order for ((n^a)-1)/a) to equal e. So, like any normal person, I went straight to Desmos and graphed ((n^x)-1)/x) and y=e, looking for what number I had to put in n to approximate the point at which they equal at around zero (which I know is undefined, but considering that