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

Worked example: Product rule with table | Derivative rules | AP Calculus AB | Khan Academy


2m read
·Nov 11, 2024

The following table lists the values of functions F and H and of their derivatives f prime and H prime for x is equal to 3. So, let's just tell us when x is equal to three, the value of the function is six. F of three is six. You could view it that way: H of three is zero. F prime of three is six and H prime of three is four.

And now they want us to evaluate the derivative with respect to x of the product of F of x and H of x when x is equal to 3. So, one way you could view this is if we viewed some function G of x as being equal to the product of F of x and H of x. This expression is the derivative of G of x. So, we could write G prime of x is equal to the derivative with respect to x of F of x times H of x, which is what we see right here and which is what we want to evaluate at x equal to 3.

So, we essentially want to evaluate G prime of three. This is what they're asking us to do. Well, to do that, let's go first up here; let's just think about what this is doing. They're asking us to take the derivative with respect to x of the product of two functions that we have some information about. Well, if we're taking the derivative of the product of two functions, you could imagine that the product rule could prove useful here.

So, I'm just going to restate the product rule. This is going to be equal to the derivative of the first function, F prime of x, times the second function—not taking its derivative—plus the first function, not taking its derivative, F of x, times the derivative of the second function, H prime of x.

So, if you're trying to find G prime of 3, well that's just going to be F prime of 3 times H of 3 plus F of 3 times H prime of three. And lucky for us, they give us what all of these things evaluate to. F prime of three right over here, they tell us F prime when x is equal to 3 is equal to 6. So this right over here is six.

H of three, they give us that too. H of three when x is three, the value of our function is zero. So this is zero. So this first term, you just get six times zero, which is going to be zero. But we'll get to that now. F of three. F of three, well the function when x is equal to 3, or the function F of three, is equal to 6, so that is six.

And then finally, the H prime evaluated at three: H prime of x when x is equal to 3, H prime of x is equal to 4, or you could say this is H prime of three. So this is four. And so, there you have it; this is going to evaluate to 6 times 0, which that’s all just going to be 0.

0 plus 6 times 4, which is going to be equal to 24, and we're done.

More Articles

View All
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…
Genes, traits, and the environment | Inheritance and variation | High school biology | Khan Academy
This is a prize-winning Himalayan rabbit, and it will help us see that an organism’s traits aren’t only the results of which genes they have, but also which environmental factors the organism is exposed to. So, we’re going to look at a specific gene in t…
Anti-Gravity Wheel?
I am here at the University of Sydney where the mechanical engineering shop has built this incredible piece of apparatus for me. It is a forty pound, that is nineteen kilogram flywheel on the end of a meter long shaft. Can you imagine trying to hold this …
The 5 Secrets to making a TON of Money
What’s up, you guys? It’s Graham Years. So, when it comes to learning how to make more money, I’ve noticed a few major differences between the people who make a normal average income and the people who end up making a ton of money. And when I talk about …
Warren Buffett's 3 New Stocks for 2022!
Well, it’s that time again. 13F season is upon us, which means we get to glimpse inside the buys themselves of the world’s, uh, biggest and most successful investors. Today, we’re kicking things off with the granddaddy himself, Mr. Warren Buffett. So, th…
I Waterproofed Myself With Aerogel!
I feel confident. “You’re confident that I am NOT gonna be damaged? Not permanently? Okay, let me back up for a moment. I want to talk about the properties of aerogel, the world’s lightest solid. What I’m gonna do is I’m gonna lean in so it’s coming in t…