yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Basic derivative rules (Part 1) | Derivative rules | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

So these are both ways that you will see limit-based definitions of derivatives. Usually, this is if you're thinking about the derivative at a point. Here, if you're thinking about the derivative in general, but these are both equivalent. They're both based on the slope of a tangent line or the instantaneous rate of change. Using these, I want to establish some of the core properties of derivatives for us.

The first one that I'm going to do will seem like common sense, or maybe it will once we talk about it a little bit. So, if F of x, if our function is equal to a constant value, well then F prime of x is going to be equal to zero. Now, why does that make intuitive sense? Well, we could graph it. We could graph it. So, if that's my y-axis, that's my x-axis. If I wanted to graph y = F of x, it's going to look like that, where this is at the value y is equal to K.

So this is y is equal to F of x. Notice, no matter what you change x, y does not change. The slope of the tangent line here, well frankly, is the same line. It has a slope of zero. No matter how y is just not changing here, we could use either of these definitions to establish that even further, establish it using these limit definitions.

So let's see the limit, and as h approaches zero of f of x + h. Well, no matter what we input into our function, we get K. So f of x plus h would be K minus F of x. Well, no matter what we put into that function, we get K over h. Well, this is just going to be 0 over h, so this limit is just going to be equal to zero.

So, f prime of x for any x, the derivative is zero. And you see that here, that this slope of the tangent line for any x is equal to zero. So, if someone walks up to you on the street and says, "Okay, h of x, h of x, h of x is equal to pi, what is h prime of x?" You say, "Well, pi, that's just a constant value. The value of our function is not changing as we change our x. The slope of the tangent line there, the instantaneous rate of change, is going to be equal to zero."

More Articles

View All
This Disease is Deadlier Than The Plague
Hello, Steve here. Today I am moving over as the voice of Kurzgesagt for something really special. Our dearest friend John Green would like to tell you a story that’s very close to his heart. So, let’s hear it from him directly. Hey, John! Hey, Steve. T…
Neil deGrasse Tyson Talks Life on Mars | StarTalk
Uh, Larry Wour had a question for me. He was—he was like totally there too. Let’s find out. So, let me ask you this: um, Mars, is there a possibility that there could have been— I don’t mean microbial life, but I mean actual intelligent, like human life …
Justification with the intermediate value theorem: table | AP Calculus AB | Khan Academy
The table gives selected values of the continuous function f. All right, fair enough. Can we use the Intermediate Value Theorem to say that the equation f of x is equal to 0 has a solution where 4 is less than or equal to x is less than or equal to 6? If …
Crypto Will Be The 12th Sector of The S&P! | Bitcoin 2022
[Music] It’s pretty chaotic here on the first day because nobody knows where to go. There’s 50,000 people showing. The first day probably about 250,000 by the time this is over, and it’s really going to be big this year because there’s so many institution…
Greenhouse effect and greenhouse gases | High school biology | Khan Academy
In this video, we’re going to talk about the greenhouse effect and also the greenhouse gases which cause the greenhouse effect. Now let’s just start with a basic idea. Imagine if Earth had no atmosphere. What would happen? Well, you have the sun, which is…
How Do You Make a Skateboard Out of Trash? | Short Film Showcase
My name is Mac Primo. I’m an artist, director, stuff maker. I was contacted by the World Wildlife Fund; it’s here for Earth Hour. They asked 15 artists from around the world to take stuff that already exists, repurpose it, give it a new life. That’s prett…