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  • Now that we know the power rule, and we saw that

  • in the last video, that the derivative with respect

  • to x, of x to the n, is going to be equal to n times x to the n

  • minus 1 for n not equal 0.

  • I thought I would expose you to a few more rules or concepts

  • or properties of derivatives that essentially will allow

  • us to take the derivative of any polynomial.

  • So this is powerful stuff going on.

  • So the first thing I want to think about

  • is, why this little special case for n not equaling 0?

  • What happens if n equals 0?

  • So let's just think of the situation.

  • Let's try to take the derivative with respect

  • to x of x to the 0 power.

  • Well, what is x to the 0 power going to be?

  • And we can assume that x for this case right over here

  • is not equal to 0.

  • 0 to the 0, weird things happen at that point.

  • But if x does not equal 0, what is

  • x to the 0 power going to be?

  • Well, this is the same thing as the derivative with respect

  • to x of 1.

  • x to the 0 power is just going to be 1.

  • And so what is the derivative with respect to x of 1?

  • And to answer that question, I'll just graph it.

  • I'll just graph f of x equals 1 to make

  • it a little bit clearer.

  • So that's my y-axis.

  • This is my x-axis.

  • And let me graph y equals 1, or f of x equals 1.

  • So that's 1 right over there. f of x

  • equals 1 is just a horizontal line.

  • So that right over there is the graph,

  • y is equal to f of x, which is equal to 1.

  • Now, remember the derivative, one way to conceptualize

  • is just the slope of the tangent line at any point.

  • So what is the slope of the tangent line at this point?

  • And actually, what's the slope at every point?

  • Well, this is a line, so the slope doesn't change.

  • It has a constant slope.

  • And it's a completely horizontal line.

  • It has a slope of 0.

  • So the slope at every point over here,

  • slope is going to be equal to 0.

  • So the slope of this line at any point

  • is just going to be equal to 0.

  • And that's actually going to be true for any constant.

  • The derivative, if I had a function, let's say that f of x

  • is equal to 3.

  • Let's say that's y is equal to 3.

  • What's the derivative of y with respect

  • to x going to be equal to?

  • And I'm intentionally showing you all the different ways

  • of the notation for derivatives.

  • So what's the derivative of y with respect to x?

  • It can also be written as y prime.

  • What's that going to be equal to?

  • Well, it's the slope at any given point.

  • And you see that no matter what x you're looking at,

  • the slope here is going to be 0.

  • So it's going to be 0.

  • So it's not just x to the 0.

  • If you take the derivative of any constant,

  • you're going to get 0.

  • So let me write that.

  • Derivative with respect to x of any constant--

  • so let's say of a where this is just a constant, that's

  • going to be equal to 0.

  • So pretty straightforward idea.

  • Now let's explore a few more properties.

  • Let's say I want to take the derivative with respect

  • to x of-- let's use the same A. Let's say

  • I have some constant times some function.

  • Well, derivatives work out quite well.

  • You can actually take this little scalar multiplier,

  • this little constant, and take it out of the derivative.

  • This is going to be equal to A. I didn't

  • want to do that magenta color.

  • It's going to be equal to A times the derivative of f of x.

  • Let me do that blue color.

  • And the other way to denote the derivative of f of x

  • is to just say that this is the same thing.

  • This is equal to A times this thing right over here

  • is the exact same thing as f prime of x.

  • Now this might all look like really fancy notation,

  • but I think if I gave you an example

  • it might make some sense.

  • So what about if I were to ask you the derivative with respect

  • to x of 2 times x to the fifth power?

  • Well, this property that I just articulated

  • says, well, this is going to be the same thing as 2

  • times the derivative of x to the fifth, 2 times the derivative

  • with respect to x of x to the fifth.

  • Essentially, I could just take this scalar multiplier

  • and put it in front of the derivative.

  • So this right here, this is the derivative

  • with respect to x of x to the fifth.

  • And we know how to do that using the power rule.

  • This is going to be equal to 2 times-- let me write that.

  • I want to keep it consistent with the colors.

  • This is going to be 2 times the derivative of x to the fifth.

  • Well, the power rule tells us, n is 5.

  • It's going to be 5x to the 5 minus 1 or 5x

  • to the fourth power.

  • So it's going to be 5x to the fourth power, which

  • is going to be equal to 2 times 5 is 10, x to the fourth.

  • So 2x to the fifth, you can literally just

  • say, OK, the power rule tells me derivative

  • of that is 5x to the fourth.

  • 5 times 2 is 10.

  • So that simplifies our life a good bit.

  • We can now, using the power rule and this one property,

  • take the derivative anything that

  • takes the form Ax to the n power.

  • Now let's think about another very useful

  • derivative property.

  • And these don't just apply to the power rule,

  • they apply to any derivative.

  • But they are especially useful for the power rule

  • because it allows us to construct polynomials and take

  • the derivatives of them.

  • But if I were to take the derivative

  • of the sum of two functions-- so the derivative of,

  • let's say one function is f of x and then

  • the other function is g of x.

  • It's lucky for us that this ends up

  • being the same thing as the derivative of f

  • of x plus the derivative of g of x.

  • So this is the same thing as f-- actually,

  • let me use that derivative operator just to make it clear.

  • It's the same thing as the derivative with respect

  • to x of f of x plus the derivative with respect

  • to x of g of x.

  • So we'll put f of x right over here

  • and put g of x right over there.

  • And so with the other notation, we

  • can say this is going to be the same thing.

  • Derivative with respect to x of f of x, we can write as f

  • prime of x.

  • And the derivative with respect to x of g of x,

  • we can write as g prime of x.

  • Now, once again, this might look like kind of fancy notation

  • to you.

  • But when you see an example, it'll make it pretty clear.

  • If I want to take the derivative with respect

  • to x of let's say x to the third power

  • plus x to the negative 4 power, this just

  • tells us that the derivative of the sum

  • is just the sum of the derivatives.

  • So we can take the derivative of this term using the power rule.

  • So it's going to be 3x squared.

  • And to that, we can add the derivative of this thing right

  • over here.

  • So it's going to be plus-- that's

  • a different shade of blue-- and over here is negative 4.

  • So it's plus negative 4 times x to the negative 4 minus 1,

  • or x to the negative 5 power.

  • So we have-- and I could just simplify a little bit.

  • This is going to be equal to 3x squared

  • minus 4x to the negative 5.

  • And so now we have all the tools we need in our toolkit

  • to essentially take the derivative of any polynomial.

  • So let's give ourselves a little practice there.

  • So let's say that I have-- and I'll do it in white.

  • Let's say that f of x is equal to 2x

  • to the third power minus 7x squared plus 3x minus 100.

  • What is f prime of x?

  • What is the derivative of f with respect to x going to be?

  • Well, we can use the properties that we just said.

  • The derivative of this is just going

  • to be 2 times the derivative of x to the third.

  • Derivative of x to the third is going to be 3x squared,

  • so it's just going to be 2 times 3x squared.

  • What's the derivative of negative 7x

  • squared going to be?

  • Well, it's just going to be negative 7 times

  • the derivative of x squared, which is 2x.

  • What is the derivative of 3x going to be?

  • Well, it's just going to be 3 times the derivative of x,

  • or 3 times the derivative of x to the first.

  • The derivative of x to the first is just 1.

  • So this is just going to be plus 3 times--

  • we could say 1x to the 0-- but that's just 1.

  • And then finally, what's the derivative of a constant

  • going to be?

  • Let me do that in a different color.

  • What's the derivative of a constant going to be?

  • Well, we covered that at the beginning of this video.

  • The derivative of any constant is just

  • going to be 0, so plus 0.

  • And so now we are ready to simplify.

  • The derivative of f is going to be

  • 2 times 3x squared is just 6x squared.

  • Negative 7 times 2x is negative 14x plus 3.

  • And we don't have to write the 0 there.

  • And we're done.

  • We now have all the properties in our tool belt

  • to find the derivative of any polynomial

  • and actually things that even go beyond polynomials.

Now that we know the power rule, and we saw that

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Differentiating polynomials | 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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