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
How We Can Keep Plastics Out of Our Ocean | National Geographic
8 million metric tons of plastic trash enters the sea from land every year; the equivalent of five plastic bags filled with trash for every foot of coastline in the world. Across our ocean, plastic trash blows into circulation, dispersed almost everywhere…
How To Supercharge Your Execution Skill
The world belongs to those who act. If everyone would do what they say they would, we’d have flying cars by now. But we don’t. The average individual has very poor execution skills. They’re blocked by a lack of discipline, a mountain of procrastination, a…
Generating Wind Power | Live Free or Die
We got a whole slew of scrap line around our property, and we happen to have a treadmill that we could probably salvage the motor from and, uh, use it for a generator. Whoa, crazy! That was nuts! That was easy! What are you doing? I’m taking this thing a…
Interpreting equations graphically | Mathematics III | High School Math | Khan Academy
Let F of x = 3x - 5 and g of x = x^3 - 4x^2 + x + 6. The graphs of y = F of x and y = G of x are shown below, and we see them right over here. This y = F of x is in, that is, in that purplish color. Let me see if I can get that same purplish color so tha…
How Secure is Your Password? And 21 Other DONGs
Hey, Vsauce. Michael here. And are you still doing things in the real world? C’mon, I mean, why flip a coin when you could just flipacoin.com? Every time you refresh the page, it flips again. Of course, there are plenty of other things you can Do Online N…
You’re reading wrong! 5 tips to become a better reader 📚
One of the common mistakes about reading is that aiming too much or focusing on numbers too much. I know reading tons and tons of books might feel amazing about yourself and even maybe feed your ego a little bit. But in those scenarios, you need to rememb…