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  • What I want to do in this video is

  • to see whether the power rule is giving us

  • results that at least seem reasonable.

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

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

  • So let's say that f of x is equal to x.

  • The power rule tells us that f prime

  • of x is going to be equal to what?

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

  • So n is implicitly 1 right over here.

  • So we bring the 1 out front.

  • 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

  • is just 1.

  • So it's just going to be equal to 1.

  • Now, does that makes conceptual sense

  • if we actually try to visualize these functions?

  • So let me actually try to graph these functions.

  • So that's my y-axis.

  • This is my x-axis.

  • And let me graph y equals x.

  • So y is equal to f of x here.

  • So y is equal to x.

  • So it looks something like that.

  • So y is equal to x.

  • Or this is f of x is equal to x, or y

  • is equal to this f of x right over there.

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

  • to not confuse you.

  • So this right over here is f of x

  • is equal to x that I graphed right over here.

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

  • And now, let me graph the derivative.

  • Let me graph f prime of x.

  • That's saying it's 1.

  • That's saying it's 1 for all x.

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

  • Is this consistent with what we know

  • about derivatives and slopes and all the rest?

  • Well, let's look at our function.

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

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

  • Or it has a constant slope of 1.

  • Slope is equal to 1 no matter what x is.

  • It's a line.

  • And for a line, the slope is constant.

  • So over here, the slope is indeed 1.

  • 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

  • would be equal to what?

  • Well, n is equal to 2.

  • 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.

  • It's going to be equal to 2x.

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

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

  • precisely.

  • Let's see how precisely I can graph it.

  • So this is the x-axis, y-axis.

  • Let me mark some stuff off here.

  • So this is 1, 2, 3, 4, 5.

  • This is 1, 2, 3, 4.

  • 1, 2, 3, 4.

  • So g of x.

  • When x is 0, it's 0.

  • When x is 1, it is 1.

  • When x is negative 1, it's 1.

  • When x is 2, it is 4.

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

  • Puts us right over there.

  • When x is negative 2, you get to 4.

  • It's a parabola.

  • You've seen this for many years.

  • I put that point a little bit too high.

  • It looks something like this.

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

  • So this might be right over here.

  • So it looks something like this.

  • It looks something like that.

  • And then, when you come over here,

  • it looks something like that.

  • It's symmetric.

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

  • So there you go.

  • 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

  • telling us a g prime of x is.

  • So g prime of x is equal to 2x.

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

  • So it looks something like that.

  • When x is equal to 1, y is equal to 2.

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

  • So it looks something like this.

  • Let me try my best to draw a straight line.

  • It looks something like this.

  • Now, does this make sense?

  • Well, if you just eyeball it really fast,

  • if you look at this point right over here--

  • and you want to think about the slope of the tangent line.

  • Let me do this in a color that pops out a little bit more.

  • So the tangent line would look something like this.

  • So it looks like it has a reasonably high negative slope.

  • Yeah.

  • It's definitely a negative slope,

  • and it's a pretty steep negative slope.

  • For x is equal to negative 2, g prime of negative 2

  • is equal to 2 times negative 2, which is equal to negative 4.

  • So this is claiming that the slope at this point-- so

  • this right over here is negative 4--

  • is saying that the slope of this point

  • is negative 4. m is equal to negative 4.

  • That looks about right.

  • It's a fairly steep negative slope.

  • Now, what happens if you go right over here

  • when x is equal to 0?

  • Well, our derivative-- if you say g prime of 0--

  • is telling us that the slope of our original function, g, at x

  • is equal to 0 is 2 times 0 is 0.

  • Well, does that make sense?

  • Well, if we go to our original parabola,

  • it does indeed make sense.

  • That's the slope of the tangent line.

  • The tangent line looks something like this.

  • We're at a minimum point.

  • We're at the vertex.

  • The slope really does look to be 0.

  • And what if you go right over here

  • to x equals 2, the slope of the tangent line?

  • Well, over here, the tangent line looks something like this.

  • It looks like a fairly steep positive slope.

  • What is our derivative telling us based on the power rule?

  • So this is essentially saying, hey,

  • tell me what the slope of the tangent line for g

  • is when x is equal to 2.

  • Well, we figured it out.

  • It's going to be 2 times x.

  • It's going to be 2 times 2, which is equal to 4.

  • It's telling us that the slope over here is 4.

  • And I'm just using m. m is often the letter

  • used to denote slope.

  • They're saying that the slope of the tangent line

  • there is 4, which seems completely, completely

  • reasonable.

  • So I encourage you to try this out yourself.

  • I encourage you to try to estimate the slopes

  • by calculating, by taking points closer

  • and closer around those points.

  • And you'll see that the power rule really

  • does give you results that actually make sense.

What I want to do in this video is

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

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    yukang920108 に公開 2022 年 07 月 12 日
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