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  • - [Voiceover] So we have the function g of x,

  • which is equal to 2/x to the third minus 1/x squared.

  • And what I wanna do in this video,

  • is I wanna find what g prime of x is

  • and then I also wanna evaluate that at x equal two,

  • so I wanna figure that out.

  • And I also wanna figure out what does

  • that evaluate to when x is equal to two?

  • So what is the slope the of the tangent line

  • to the graph of g, when x is equal to two?

  • And like always, pause this video and

  • see if you can work this out on your own

  • before I work through it with you.

  • And I'll give you some hints

  • all you really need to do is

  • apply the power rule, a little bit of

  • basic exponent properties

  • and some basic derivative properties

  • to be able to do this.

  • Alright, now let's just do this together

  • and I'll just rewrite it.

  • G of x is equal to this first term here,

  • 2/x to the third.

  • Well, that could be rewritten as

  • 2 times x to the negative three.

  • We know that 1/x to the n is the same thing

  • as x to the negative n.

  • So I just rewrote it and this might

  • be ringing a bell of how the power rule

  • might be useful.

  • And then we have minus-

  • well, 1/x squared that is the same thing

  • as x to the negative two.

  • And so this, if we're gonna take

  • the derivative of both sides of this,

  • let's do that.

  • Derivative with respect to x.

  • Dx, we're gonna do that on the left-hand side,

  • we're also gonna do it on the right-hand side.

  • On the left-hand side, the derivative

  • with respect to x of g of x,

  • we can write that as g prime of x

  • is going to be equal to,

  • well, the derivative of this first

  • that we have right here written in green,

  • this is going to be, we're just gonna

  • apply the power rule.

  • We're going to take our exponent,

  • multiply it by our coefficient out front,

  • actually let me write that out,

  • that's going to be...

  • There's this equal sign.

  • That is going to be two times negative three,

  • times x and now we're going to decrement this exponent.

  • You have to be very careful here

  • because sometimes your brain might say,

  • "Okay, one less than three is two,

  • "so maybe this is x in the negative two."

  • but remember, you're going down.

  • So if you're at negative three and you subtract one,

  • we're gonna go with the negative three

  • minus one power.

  • We'll that's gonna take us to negative four.

  • So this is x to the negative four power.

  • So two times negative three x to the negative four,

  • or we could have also written that as

  • negative six x to the negative four power.

  • And then, minus...

  • Well, we're gonna do the same thing again

  • right over here.

  • We take this negative two, multiply it by

  • the coefficient that's implicitly here,

  • you could say there's a one there.

  • So negative two times one.

  • So you have the negative two there

  • and then you have the x to the-

  • well what's a negative two minus one?

  • That's negative three.

  • To the negative three power.

  • And so we can rewrite all of this business as,

  • the derivative g prime of x,

  • is equal to negative six,

  • negative six x to the negative fourth.

  • And now we're subtracting a negative.

  • So we could just write this as,

  • plus two x to the negative three.

  • This negative cancels out with that negative.

  • Subtract a negative, same thing as adding a positive.

  • So we did the first part.

  • We can express g prime of x as a function of x.

  • Now, let's just evaluate what g prime of two is.

  • So g prime of two is going to be equal to

  • negative six times two to the negative fourth power

  • plus two times two to the negative third power.

  • Well, what's this going to be?

  • This is equal to negative 6/2 to the fourth,

  • plus 2/2 to the third,

  • which is equal to negative six over-

  • two to the fourth is 16,

  • plus 2/2 to the third is eight.

  • And so let's see, this is...

  • Lets rewrite this all with a common denominator.

  • I could write this as 1/4,

  • but then this one won't work out as cleanly,

  • I could write them both as eights.

  • This is negative 3/8s.

  • Negative 3/8s.

  • So you have negative 3/8s plus two eights

  • is equal to negative 1/8.

  • So the slope of the tangent line at x equals two,

  • to the graph y equals g of x has a slope.

  • That slope is negative 1/8.

- [Voiceover] So we have the function g of x,

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A2 初級

Negative powers differentiation | 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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