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

Finding derivative with fundamental theorem of calculus: chain rule | AP®︎ Calculus | Khan Academy


2m read
·Nov 10, 2024

Let's say that we have the function capital F of x, which we're going to define as the definite integral from 1 to sine of x. So that's an interesting upper bound right over there of 2t minus 1, and of course dt.

What we are curious about is trying to figure out what is F prime of x going to be equal to. So pause this video and see if you can figure that out.

All right, so some of you might have been a little bit challenged by this notion of, hey, instead of an x on this upper bound, I now have a sine of x. If it was just an x, I could have used the fundamental theorem of calculus. Just to review that, if I had a function, let me call it h of x, if I have h of x that was defined as the definite integral from 1 to x of 2t minus 1 dt, we know from the fundamental theorem of calculus that h prime of x would be simply this inner function with the t replaced by the x. It would just be 2x minus 1. Pretty straightforward.

But this one isn't quite as straightforward. Instead of having an x up here, our upper bound is a sine of x. So one way to think about it is if we were to define g of x as being equal to sine of x, our capital F of x can be expressed as capital F of x is the same thing as h of, instead of an x everywhere we see an x, we're replacing it with a sine of x. So it's h of g of x. You can see the g of x right over there, so you replace x with g of x for where in this expression you get h of g of x, and that is capital F of x.

Now, why am I doing all of that? Well, this might start making you think about the chain rule because if this is true, then that means that capital F prime of x is going to be equal to h prime of g of x times g prime of x.

And so, what would that be? Well, we already know what h prime of x is, so let me do this in another color. This part right over here is going to be equal to everywhere we see an x here, we'll replace with the g of x. So it's going to be 2 times sine of x, and then minus 1. This is this right over here.

And then what's g prime of x? g prime of x, well, g prime of x is just, of course, the derivative of sine of x, which is cosine of x. So this part right over here is going to be cosine of x.

And we could keep going, we could try to simplify this a little bit or rewrite it in different ways, but there you have it.

More Articles

View All
Enzyme reaction velocity and pH | Cellular energetics | AP Biology | Khan Academy
In this video, we’re going to talk about enzymes. In particular, we’re going to talk about the effect of pH on enzymes—how acidic or basic the environment is and how that affects enzyme activity. So just as a bit of review, enzymes are molecules that hel…
How covid impacted private aviation! Parts-1
What do you think Covet did for the private aviation industry? Because I’ll be honest, when that whole thing was going on, that was kind of my first introduction to starting the charter travel. It got very crazy, and even though prices were quite crazy at…
Misconceptions About Falling Objects
Now I want you to make a prediction: in my left hand I have a standard size basketball, and in my right hand a 5 kg medicine ball. If I drop them both at exactly the same time, which one will hit the ground first? Ah, this is a trick one, isn’t it? The h…
My 3 Step Guide to Quickly Screen Stocks
Hey guys, welcome back to the channel. In this video, we’re going to be talking about a simple three-step method that I personally use in order to quickly screen stocks. So, we’re not thinking about, you know, potentially making an investment just yet. W…
What are affixes? | Reading | Khan Academy
Hello readers! Today we’re going to talk about things called affixes. One of the things that I love about the English language is how flexible its words can be. You can take little word parts and stick them together to make new words. If I read something…
What If You Fall into a Black Hole?
Black holes are the most powerful and extreme things in the universe, and they’re wildly weird and complicated. What would happen if you fell inside one, and what are they? [Music] Really, first we need to talk about space and time. Space and time are t…