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

Applying the chain rule twice | Advanced derivatives | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

Let's say that y is equal to sine of x squared to the third power, which of course we could also write as sine of x squared to the third power. What we're curious about is what is the derivative of this with respect to x? What is dy/dx, which we could also write as y prime?

Well, there's a couple of ways to think about it. This isn't a straightforward expression here, but you might notice that I have something being raised to the third power. In fact, if we look at the outside of this expression, we have some business in here and it's being raised to the third power.

One way to tackle this is to apply the chain rule. So, if we apply the chain rule, it's going to be the derivative of the outside with respect to the inside, or the something to the third power. The derivative of the something to the third power with respect to that something is going to be 3 times that something squared times the derivative with respect to x of that something. In this case, the something is sine.

Let me write that in blue color. It is sine of x squared. It is sine of x squared! No matter what was inside of these orange parentheses, I would put it inside of the orange parentheses and these orange brackets right over here. We learned that in the chain rule, so let's see.

We know this is just a matter of algebraic simplification, but the second part we need to now take the derivative of sine of x squared. Well, now we would want to use the chain rule again. So, I'm going to take the derivative. It’s sine of something, so this is going to be the derivative of this is going to be the sine of something with respect to something.

That is cosine of that something times the derivative with respect to x of the something. In this case, the something is x squared. And of course, we have all of this out front, which is the 3 times sine of x squared, and I could write it like this squared.

All right, so we're getting close. Now we just have to figure out the derivative with respect to x of x squared. We've seen that many times before; we just use the power rule. That's going to be 2x.

So if we wanted to write the dy/dx, we get a little bit of a mini drum roll here. This didn't take us too long! dy/dx—I'll multiply the 3 times the 2x, which is going to be 6x.

So I covered those so far times sine squared of x squared times cosine of x squared, and we are done with applying the chain rule multiple times!

More Articles

View All
Wealth and Happiness: How to Achieve Both
They lied to you. Since you were a child, you kept hearing it over and over and over again. You always have to make a choice: money or happiness. You simply can’t have both. But what if we told you that’s just a coping mechanism for poor and middle-class …
The Unintended Consequences of Playing God
Imagine you’re going blind. The world slowly becomes a blur. You can no longer see your family or your friends. You can’t see the beauty of a mountain landscape or the ripples in the ocean. Then a YouTuber comes around, offering to give you the gift of si…
The Next Stock Market Crash (How To Profit)
What’s up you guys? It’s Graham here. And just when you thought things were going well, everything gets okay. In all seriousness, we need to address a topic that not a lot of people want to think about, and that’s the fact that at some point in the future…
How Many Calories are on a Smudgy Screen?
Hey, Vsauce. Michael here. And I’m home for the holidays. I’m in my parents’ basement, using a different camera than usual. But you know what is always different? Fingerprints. The palms of our hands and the soles of our feet are weird. They are covered w…
He Grew Up on the Streets, Now He's Making Them a Better Place | Short Film Showcase
You know you can’t change the world; you have to start with yourself. I was going down a one-way street, going backwards, and I left the house. I had my gun on my hip. I kept a blunt halfway lit, had my tennis shoes tied tight. These guys, I had to jump o…
A Small Light | In Production Piece | National Geographic
Man: –take one. Director: Ready? And, action. Susanna Fogel: People tend to think that they know history, especially with very famous stories like Anne Frank’s story. But with Miep, you’re coming at it from a sideways angle that forces you to see it from…