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### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 5: Derivative as a limit

# Derivative as a limit: numerical

Sal relates the derivative of a function and the average rate of change of the function over intervals that become infinitely small.

## Want to join the conversation?

• Shouldn't the formula being used be reversed? We are going from 1.9 to 2 on the X for example, therefore wouldn't it be f(2) - f(1.9)/ 2 - 1.9? I know it's the same result either way but it seems a less clear set up.
• Like you said, either way works. I usually do it (y2)-(y1) / (x2)-(x1) with x2 being greater than x1
• so, how would i go about a numerical derivative question not set up like this? more specifically, how would you find f'(x), f'(1), etc of the function f(x) = 3x - 7 ?
(1 vote)
• It has no curve, there would be no point in finding the derivative, as it is the derivative of itself. No matter where you go on that function, the slope will always be the same.

I'm still studying too so don't take anything I say as a fact :)

(Update) Just realized that derivative is a slope so the derivative of that function would be 3 by the way.
• Does stuff within this function even matter? Or can we just assume that the limit as x approaches 2 of some function is 4?
• The limit is 4 for this function, not for any function at 2. Eg, for the function f(x)=15, the limit of f(x) at 2 is 15.
• @ Does this mean that the slope of the tangent line at of x=2 is also 4?
• How can I calculate the derivative g(x) = csc(x) as a limit. I got stuck at ((1/senxcosh+senhcosx)-(1/sin(x)))/h
(1 vote)
• This is very tough to do algebraically. The problem gets a lot easier if you look at it on a graph.
(1 vote)
• Why do the order of those intervals change?

I see it as from the first point (x,y), where y is 2 and then the second point (y,x), where y is 2. I don't get this. And in the table, why do the interval switch side when going from 1.999 to 2.001?
(1 vote)
• We want to calculate the derivative of f(x)=x^2+1 when x=2, so we use the formula for the derivative at two points which is the limit of (f(b)-f(a))/(b-a) as b approaches a from both sides. In this case he is using x for b and 2 for a so we can evaluate the derivative as a function of x as x gets closer and closer to 2 from both sides. So the table starts with x=1.9 and the interval is of x is [1.9,2] so we have (f(1.9)-f(2))/(1.9-2) which equals 3.9. The table gives you the values of (f(x)-f(2))/(x-2) for valiues of x getting closer to 2 such as 1.99, 1.999, 2.001, 2.01 and 2.1. Hope this helps.
(1 vote)
• how are definitions of derivative and limit same?
(1 vote)
• They are not the same. A derivative can be defined as a certain limit, specifically the limit as ℎ → 0 of the difference quotient:
[𝑓(𝑥 + ℎ) – 𝑓(𝑥)]/ℎ
(1 vote)
• is the formula given with the question? How was it formulated?
(1 vote)
• This might be a stupid question, but I have not been able to understand what the relation between d(x) and f'(x) is, are they the same?
(1 vote)
• d/dx [ f(x) ] is another way of writing f'(x). They're two different notations for the derivative (the first is Leibniz notation, the second is Lagrange notation). For more information on different ways to notate derivatives, you can check out this page on Wikipedia: https://en.wikipedia.org/wiki/Notation_for_differentiation .
(1 vote)

## Video transcript

- [Voiceover] Stacy wants to find the derivative of f of x equals x squared plus one at the point x equals two. Her table below shows the average rate of change of f over the intervals from x to two or from two to x, and these are closed intervals, for x-values that get increasingly closer to two. so they get-- so we're talking about the average rate of change of f over these closed intervals for x-values that get increasingly closer to two. It looks like we're going to be dealing with some type of a limit, or we're trying to calculate some type of a limit, or approximate some type of a limit. Let's read this data here. These are the x values and she's trying to find the average rate of change between each of these x-values and two, or the average rate of change of the function between when x is-- one of these x values and two, and then she has the average rate of change that she precalculated, so we don't have to get a calculator out or anything like that, and just as a reminder, how did she calculate this 3.9? Well, they tell us. She took f of 1.9, what does the function equal when x is 1.9? From that, she subtracted what is the value of the function when f is equal to two, so that's really our change in f, and she divided it by the x, which is 1.9, minus two, so change in f over change in x. What is the average rate of change of our function over that interval? She did it between 1.9 and two, she got 3.9. Then she gets closer to two, so now she's doing it between 1.99 and two and it becomes 3.99, it looks like it's getting closer to four. She gets even closer to two and the average rate of change gets even closer to four, and then she goes on the other side of two, you could view it as this is approaching, this is-- this is approaching-- this is x approaching two from the left hand side, and this is x approaching two from the right hand side. When it's 2.1, the average rate of change is 4.1. When it's 2.01, once again, we're getting closer to two, we're getting closer to two, the average rate of change is getting closer to four. The closer we get to two, the closer the average rate of change gets to four. What this data is really helping us approximate it's really saying, okay, the average rate of change we know is f of x minus f of two, over x minus two, but what we're really thinking about is, well what is the limit as x approaches two right over here? That's what this data is helping us to get at, and it looks like this limit is equal to four. They give us it in here, it says, "Look, the closer that x gets to two "from either the left hand side or the right hand side, "the closer that this expression right over here, "which is this number, gets to four." You might recognize, this is one of the definitions of a derivative. This is one of the definitions of a derivative. This right over here would be f prime of two, the derivative at x equals two is equal to the limit as x approaches two of all of this business. There's other ways to express a derivative as a limit but this is one of them. There you go, from the table, what does the derivative of f of x equals x squared plus one at x equals two appear to be? Well, the derivative at x equals two appears to be equal to four, and we're done.