Name: 01.5（Worked example: Derivative of ÃÂ(3x_-x) using the chain rule | AP Calculus AB | Khan Academy）
Uploaded: 2022-09-09T14:56:22.000Z
Duration: 5 min 31 s
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What I want to do in this video is start with the abstract--

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

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and then learn to apply it in the concrete setting.

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

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

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

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

Let's say we were trying to take the derivative

of the square root of 3x squared minus 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

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.

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

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

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.

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

Well, wherever in the derivative we saw an x,

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

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

What we're trying to solve right over here,

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

So the derivative of f of the outer function with respect

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

This is exactly this based on how we've defined

at this, the derivative of the outer thing,

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

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

that something to the negative 1/2 power.

But now we have to take the derivative of our something

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

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

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

is this right over here and we're multiplying.

of the outer function with respect to the inner.

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

times the derivative of the inner function with respect

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01.5（Worked example: Derivative of ÃÂ(3x_-x) using the chain rule | AP Calculus AB | Khan Academy）

essentially

figure

expression

negative

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

essentially

figure

expression

negative

01.5（Worked example: Derivative of ÃÂ(3x_-x) using the chain rule | AP Calculus AB | Khan Academy）