 ## Subtitles section Play video

• What I want to do in this video is start with the abstract--

• actually, let me call it formula for the chain rule,

• and then learn to apply it in the concrete setting.

• So let's start off with some function,

• some expression that could be expressed

• as the composition of two functions.

• So it can be expressed as f of g of x.

• So it's a function that can be expressed as a composition

• or expression that can be expressed

• as a composition of two functions.

• Let me get that same color.

• I want the colors to be accurate.

• And my goal is to take the derivative of this business,

• the derivative with respect to x.

• And what the chain rule tells us is

• that this is going to be equal to the derivative

• of the outer function with respect to the inner function.

• And we can write that as f prime of not x, but f prime

• of g of x, of the inner function.

• f prime of g of x times the derivative

• of the inner function with respect to x.

• Now this might seem all very abstract and math-y.

• How do you actually apply it?

• Well, let's try it with a real example.

• Let's say we were trying to take the derivative

• of the square root of 3x squared minus x.

• So how could we define an f and a g

• so this really is the composition

• of f of x and g of x?

• Well, we could define f of x.

• If we defined f of x as being equal to the square root of x,

• and if we defined g of x as being equal to 3x squared

• minus x, then what is f of g of x?

• Well, f of g of x is going to be equal to-- I'm

• going to try to keep all the colors accurate,

• hopefully it'll help for the understanding.

• f of g of x is equal to-- where everywhere you see the x,

• you replace with the g of x-- the principal root of g of x,

• which is equal to the principal root of-- we

• defined g of x right over here-- 3x squared minus x.

• So this thing right over here is exactly

• f of g of x if we define f of x in this way

• and g of x in this way.

• Fair enough.

• So let's apply the chain rule.

• What is f prime of g of x going to be equal to,

• the derivative of f with respect to g?

• Well, what's f prime of x?

• f prime of x is equal to-- this is the same thing

• as x to the 1/2 power, so we can just apply the power rule.

• So it's going to be 1/2 times x to the--

• and then we just take 1 away from the exponent, 1/2 minus 1

• is negative 1/2.

• And so what is f prime of g of x?

• Well, wherever in the derivative we saw an x,

• we can replace it with a g of x.

• So it's going to be 1/2 times-- instead

• of an x to the negative 1/2, we can write a g of x to the 1/2.

• And this is just going to be equal to-- let

• me write it right over here.

• It's going to be equal to 1/2 times

• all of this business to the negative 1/2 power.

• So 3x squared minus x, which is exactly what we

• need to solve right over here. f prime of g of x

• is equal to this.

• So this part right over here I will--

• let me square it off in green.

• What we're trying to solve right over here,

• f prime of g of x, we've just figured out

• is exactly this thing right over here.

• So the derivative of f of the outer function with respect

• to the inner function.

• So let me write it.

• It is equal to 1/2 times g of x to the negative 1/2,

• times 3x squared minus x.

• This is exactly this based on how we've defined

• f of x and how we've defined g of x.

• Conceptually, if you're just looking

• at this, the derivative of the outer thing,

• you're taking something to the 1/2 power.

• So the derivative of that whole thing

• with respect to your something is going to be 1/2 times

• that something to the negative 1/2 power.

• That's essentially what we're saying.

• But now we have to take the derivative of our something

• with respect to x.

• And that's more straightforward.

• g prime of x-- we just use the power rule for each

• of these terms-- is equal to 6x to the first,

• or just 6x minus 1.

• So this part right over here is just going to be 6x minus 1.

• Just to be clear, this right over here

• is this right over here and we're multiplying.

• And we're done.

• We have just applied the power rule.

• So just to review, it's the derivative

• of the outer function with respect to the inner.

• So instead of having 1/2x to the negative 1/2,

• it's 1/2 g of x to the negative 1/2,

• times the derivative of the inner function with respect

• to x, times the derivative of g with respect

• to x, which is right over there.

What I want to do in this video is start with the abstract--

Subtitles and vocabulary

Click the word to look it up Click the word to find further inforamtion about it

A2 US derivative equal respect prime rule squared

# Worked example: Derivative of ÃÂ(3x_-x) using the chain rule | AP Calculus AB | Khan Academy

• 2 0
yukang920108 posted on 2022/09/09
Video vocabulary