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I'm trying to understand differentiation. At 9:45 the kind sir is filling in numbers in this video.

My question is, where is he getting the gradient numbers from when hes filling in the table? I know it can be calculated by using nx^n-1, but I don't think hes using it over there because he only explains this at the time the video reaches 13:09.

Also can someone tell me when differentiating applies? Is it just in parabolic shapes? (I dont feel like its worth opening a new question for this)

Git Gud
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user1534664
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1 Answers1

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Well, that method you've mentioned is probably how he's been able to write them down so quickly, but he just finished explaining how if you take $\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}$, and let $\Delta x=0.1,0.01,0.001,\ldots$, then you get a good idea of where things are headed.

Basically, a function (of one variable) is differentiable at a point if its graph has a tangent line at that point, which happens when the graph is 'smooth' there.

Speaking in precise terms, though, a function defined in a small interval centered at a point $x_0$ (and perhaps elsewhere, though that is of no consequence here) is said to be differentiable there if there exists $L\in\mathbb{R}$ such that for every converging sequence $x_n\to x_0$ it holds that $|\frac{f(x_n)-f(x_0)}{x_n-x_0}-L|\to 0$. We then have $y = f(x_0) + L(x-x_0)$ as the tangent to the function's graph at $(x_0,f(x_0))$.

Jonathan Y.
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  • Thanks for your answer. When hes filling in the table hes also filling in x=2 and x=1, but what I understand is that you need two points to calculate the tangent/slope. At 4:30 he shows an example of this. But like I asked in my question he just fills in the table with nx^n-1? That doesn't make sense to me, because I think hes filling in the tables to show where he got the gradient function nx^n-1 from, so why would he be using it first and then show where he gets it from? – user1534664 Aug 10 '13 at 00:27
  • I dont understand all the symbols yet, I think I'm going to have to get better before I can get to that. what is x with indice 0? – user1534664 Aug 10 '13 at 00:30

  • Look from around 4:00. He estimates the slope from $3$ to $3.2$, then $3.1$, then $3.05$, and so on. With this process he's able to 'estimate' what the gradient ($L$, in our noation) is going to be. Later on he's 'prepared' the results for similar cases 'in advance', although what he's really doing is using the formula he later establishes, of course.

  • $x_0$ is just a fixed point (zero is a nice index for that purpose) in the domain of the function. The main thing to know here is this: are you familiar with the notion of sequences and their limits?

    • – Jonathan Y. Aug 10 '13 at 00:34
  • Since you need two points before u can calculate the slope, can you just use any random point that is on the parabolic? For example at 4:41 the sir keeps using point p(3,9) to calculate delta x and delta y. What point would he use for when x is 2 later in the video when he fills in the table? – user1534664 Aug 10 '13 at 00:46
  • what he did was use $(3,9),(3.2,10.24)$, then use $(3,9),(3.1,9.61)$, then $(3,9),(3.05,9.3025)$, and so on. The corresponding slopes would be $6.2,6.1,6.05,\ldots$ – Jonathan Y. Aug 10 '13 at 00:50
  • Again, thanks alot for your help! I'm still a bit confused. At around 5:00, he is calculating the slope of line PR, but what is he calculating when hes filling in the table at around 7:30? – user1534664 Aug 10 '13 at 01:01
  • He's doing the same thing he did before, but without showing us the process. I.e. approximating using pairs of points closer and closer together. – Jonathan Y. Aug 10 '13 at 01:04
  • I finally get it now. For some reason I couldn't grasp my mind on it, maybe because its 3:30 am. Last question: So basically, once he filled the tables he can then derive the formula nx^n-1 from the patterns inside the table answers? Is that correct? – user1534664 Aug 10 '13 at 01:27