Name: Justifying the power rule | Derivative rules | AP Calculus AB | Khan Academy
Uploaded: 2022-07-12T12:46:47.000Z
Duration: 6 min 42 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

to see whether the power rule is giving us

This is by no means a proof of the power rule,

but at least we'll feel a little bit more comfortable using it.

Well, x is the same thing as x to the first power.

It'll be 1 times x to the 1 minus 1 power.

So it's going to be 1 times x to the 0 power. x to the 0

if we actually try to visualize these functions?

So let me actually try to graph these functions.

is equal to this f of x right over there.

Now, actually, let me just call that f of x just

is equal to x that I graphed right over here.

y is equal to f of x, which is equal to x.

Regardless of what x is, it's going to be equal to 1.

about derivatives and slopes and all the rest?

What is the slope of the tangent line right at this point?

Well, right over here, this has slope 1 continuously.

If you go to this point over here, the slope is indeed 1.

If you go over here, the slope is indeed 1.

So we've got a pretty valid response there.

Now, let's try something where the slope might change.

So let's say I have g of x is equal to x squared.

The power rule tells us that g prime of x

So it's going to be 2 times x to the 2 minus 1.

Or it's going to be equal to 2 x to the first power.

So let's see if this makes a reasonable sense.

And I'm going to try to graph this one a little bit more

So that puts us right over there-- 1, 2, 3, 4.

Actually, the last two points I graphed are a little bit weird.

So I'm trying my best to draw it reasonably.

That's the graph of g of x. g of x is equal to x squared.

Now, let's graph g prime of x or what the power rule is

So that's just a line that goes through the origin of slope 2.

When x is equal to 2, y or g of x is equal to 4.

Let me try my best to draw a straight line.

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Justifying the power rule | Derivative rules | AP Calculus AB | Khan Academy

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weird

essentially

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