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If you were to draw a secant line between these two points, we essentially just calculated the slope of that secant line. And so the average rate of change between two points, that is the same thing as the slope of the secant line. And by looking at the secant line in comparison to the curve over that interval, it hope... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Because in the beginning part of the interval, you see that the secant line is actually increasing at a faster rate. But then as we get closer to three, it looks like our yellow curve is increasing at a faster rate than the secant line, and then they eventually catch up. And so that's why the slope of the secant line i... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Is it the exact rate of change at every point? Absolutely not. The curve's rate of change is constantly changing. It's at a slower rate of change in the beginning part of this interval, and then it's actually increasing at a higher rate as we get closer and closer to three. So over the interval, their change in y over ... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
It's at a slower rate of change in the beginning part of this interval, and then it's actually increasing at a higher rate as we get closer and closer to three. So over the interval, their change in y over the change in x is exactly the same. Now one question you might be wondering is, why are you learning this in a ca... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
Couldn't you have learned this in an algebra class? The answer is yes. But what's going to be interesting, and it's really one of the foundational ideas of calculus, is, well, what happens as these points get closer and closer together? We found the average rate of change between one and three, or the slope of the seca... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
We found the average rate of change between one and three, or the slope of the secant line from one comma one to three comma nine. But what instead if you found the slope of the secant line between two comma four and three comma nine? So what if you found this slope? But what if you wanted to get even closer? Let's say... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
But what if you wanted to get even closer? Let's say you wanted to find the slope of the secant line between the point 2.5, 6.25, and three comma nine. And what if you just kept getting closer and closer and closer? Well then, the slopes of these secant lines are gonna get closer and closer to the slope of the tangent ... | Secant lines & average rate of change Derivatives introduction AP Calculus AB Khan Academy.mp3 |
So I've got a 10-foot ladder that's leaning against a wall, but it's on very slick ground, and it starts to slide outward. And right at the moment that we're looking at this ladder, the base of the ladder is 8 feet away from the base of the wall, and it's sliding outward at 4 feet per second. And we'll assume that the ... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
It stays kind of in contact with the wall and moves straight down. And we see right over here, the arrow is moving straight down. And our question is, how fast is it moving straight down at that moment? So let's think about this a little bit. What do we know and what do we not know? So if we call the distance between t... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
So let's think about this a little bit. What do we know and what do we not know? So if we call the distance between the base of the wall and the base of the ladder, let's call that x. We know right now x is equal to 8 feet. We also know the rate at which x is changing with respect to time. The rate at which x is changi... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
We know right now x is equal to 8 feet. We also know the rate at which x is changing with respect to time. The rate at which x is changing with respect to time is 4 feet per second. So we could call this dx dt. Now let's call the distance between the top of the ladder and the base of the ladder h. Let's call that h. So... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
So we could call this dx dt. Now let's call the distance between the top of the ladder and the base of the ladder h. Let's call that h. So what we're really trying to figure out is what dh dt is, given that we know all of this other information. So let's see if we can come up with a relationship between x and h, and th... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
Well, we know the relationship between x and h at any time because of the Pythagorean theorem. We can assume this is a right angle. So we know that x squared plus h squared is going to be equal to the length of the ladder squared, is going to be equal to 100. And what we care about is the rate at which these things cha... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
And what we care about is the rate at which these things change with respect to time. So let's take the derivative with respect to time of both sides of this. So we're doing a little bit of implicit differentiation. So what's the derivative with respect to time of x squared? Well, the derivative of x squared with respe... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
So what's the derivative with respect to time of x squared? Well, the derivative of x squared with respect to x is 2x. And we're going to have to multiply that times the derivative of x with respect to t, dx dt. Just to be clear, this is a chain rule. This is the derivative of x squared with respect to x, which is 2x, ... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
Just to be clear, this is a chain rule. This is the derivative of x squared with respect to x, which is 2x, times dx dt to get the derivative of x squared with respect to time, just the chain rule. Now similarly, what's the derivative of h squared with respect to time? Well, that's just going to be 2h. The derivative o... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
Well, that's just going to be 2h. The derivative of h squared with respect to h is 2h times the derivative of h with respect to time. Once again, this right over here is the derivative of h squared with respect to h times the derivative of h with respect to time, which gives us the derivative of h squared with respect ... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
And what do we get on the right-hand side of our equation? Well, the length of our ladder isn't changing. This 100 isn't going to change with respect to time. The derivative of a constant is just equal 0. So now we have it, a relationship between the rate of change of h with respect to time, the rate of change of x wit... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
The derivative of a constant is just equal 0. So now we have it, a relationship between the rate of change of h with respect to time, the rate of change of x with respect to time, and then at a given point in time when the length of x is x and h is h. But do we know what h is when x is equal to 8 feet? Well, we can fig... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
When x is equal to 8 feet, we can use the Pythagorean theorem again. We get 8 feet squared plus h squared is going to be equal to 100. So 8 squared is 64. Subtract it from both sides. You get 8 squared is equal to 36. Take the positive square root. A negative square root doesn't make sense because then the ladder would... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
Subtract it from both sides. You get 8 squared is equal to 36. Take the positive square root. A negative square root doesn't make sense because then the ladder would be below the ground. It would be somehow underground. So we get h is equal to 6. So this is something that was essentially given by the problem. | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
A negative square root doesn't make sense because then the ladder would be below the ground. It would be somehow underground. So we get h is equal to 6. So this is something that was essentially given by the problem. So now we know. We can look at this original thing right over here. We know what x is. | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
So this is something that was essentially given by the problem. So now we know. We can look at this original thing right over here. We know what x is. That was given. Right now, x is 8 feet. We know the rate of change of x with respect to time. | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
We know what x is. That was given. Right now, x is 8 feet. We know the rate of change of x with respect to time. It's 4 feet per second. We know what h is right now. It is 6. | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
We know the rate of change of x with respect to time. It's 4 feet per second. We know what h is right now. It is 6. So then we can solve for the rate of h with respect to time. So let's do that. So we get 2 times 8 feet times 4 feet per second plus 2h. | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
It is 6. So then we can solve for the rate of h with respect to time. So let's do that. So we get 2 times 8 feet times 4 feet per second plus 2h. So plus 2h is going to be plus 2 times our height right now is 6 times the rate at which our height is changing with respect to t is equal to 0. And so we get 2 times 8 times... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
So we get 2 times 8 feet times 4 feet per second plus 2h. So plus 2h is going to be plus 2 times our height right now is 6 times the rate at which our height is changing with respect to t is equal to 0. And so we get 2 times 8 times 4 is 64 plus 12 dh dt is equal to 0. We can subtract 64 from both sides. We get 12 time... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
We can subtract 64 from both sides. We get 12 times the derivative of h with respect to time is equal to negative 64. And then we just have to divide both sides by 12. And so now we get a little bit of a drum roll. The derivative, the rate of change of h with respect to time is equal to negative 64 divided by 12 is equ... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
And so now we get a little bit of a drum roll. The derivative, the rate of change of h with respect to time is equal to negative 64 divided by 12 is equal to negative 64 over 12, which is the same thing as negative 16 over 3. Yep, that's right, which is equal to negative 5 and 1 third feet per second. So we're done. Bu... | Related rates Falling ladder Applications of derivatives AP Calculus AB Khan Academy.mp3 |
We've already seen one definition of the definite integral, and many of them are closely related to this definition that we've already seen, is, well, look, the definite integral from a to b of f of x, d of x is this area shaded in blue, and we can approximate it by splitting it into n rectangles, so let's say that's t... | Switching bounds of definite integral AP Calculus AB Khan Academy.mp3 |
How do you think these two things should relate? And I encourage you to look at all of this to come to that conclusion, and pause the video to do so. Well, let's just think about what's going to happen. This is going to be, if I were to literally just take this, if I were to literally just take this and copy and paste ... | Switching bounds of definite integral AP Calculus AB Khan Academy.mp3 |
This is going to be, if I were to literally just take this, if I were to literally just take this and copy and paste it, which is exactly what I'm going to do, if I just took this, by definition, since I swapped these two bounds, I am going to want to swap these two. Instead of b minus a, it's going to be a minus b now... | Switching bounds of definite integral AP Calculus AB Khan Academy.mp3 |
So each of these are going, this value right over here, let me make these color-coded maybe, so this orange delta x, this orange delta x is going to be the negative of this green, of this green delta x. This is the negative of that right over there. And everything else is the same. So what am I going to end up doing? W... | Switching bounds of definite integral AP Calculus AB Khan Academy.mp3 |
So what am I going to end up doing? Well, I'm essentially going to end up having the negative value of this. So this is going to be equal to the negative of the integral from a to b of f of x dx. And so this is the result we get, which is another really important integration property that if you swap, if you swap the b... | Switching bounds of definite integral AP Calculus AB Khan Academy.mp3 |
So one way you could think of it, if you set f of x as being equal to sine of x, and g of x being the natural log of x, and let's see, fg, let's say h of x, h of x equaling x squared, then this thing right over here is the exact same thing as trying to take the derivative with respect to x of f of g of h of x. And what... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
So the derivative of sine of x is cosine of x. But instead of it being cosine of x, it's going to be cosine of whatever was inside of here. So it's going to be cosine of natural log. Let me write that in that same color. Cosine of natural log of x squared. I'm going to do x in that same yellow color. Cosine of x square... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
Let me write that in that same color. Cosine of natural log of x squared. I'm going to do x in that same yellow color. Cosine of x squared. And so you could really view this part, what I just write over here, as f prime. This is f prime of g of h of x. This is f prime of g of h of x. | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
Cosine of x squared. And so you could really view this part, what I just write over here, as f prime. This is f prime of g of h of x. This is f prime of g of h of x. If you want to keep track of things. So I just took the derivative of the outer with respect to whatever was inside of it. And now I have to take the deri... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
This is f prime of g of h of x. If you want to keep track of things. So I just took the derivative of the outer with respect to whatever was inside of it. And now I have to take the derivative of the inside with respect to x. But now we have another composite function. So we're going to multiply this times. We're going... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And now I have to take the derivative of the inside with respect to x. But now we have another composite function. So we're going to multiply this times. We're going to do the chain rule again. The derivative of, we're going to take the derivative of ln with respect to x squared. So the derivative of ln of x is 1 over ... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
We're going to do the chain rule again. The derivative of, we're going to take the derivative of ln with respect to x squared. So the derivative of ln of x is 1 over x. But now we're going to have 1 over not x, but 1 over x squared. So to be clear, this part right over here is g prime of not x. If it was g prime of x, ... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
But now we're going to have 1 over not x, but 1 over x squared. So to be clear, this part right over here is g prime of not x. If it was g prime of x, this would be 1 over x. But instead of an x, we have our h of x there. We have our x squared. So it's g prime of x squared. And then finally, we can take the derivative ... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
But instead of an x, we have our h of x there. We have our x squared. So it's g prime of x squared. And then finally, we can take the derivative of our inner function. Let me write it. So we could write this as g prime of h of x. And finally, we just have to take the derivative of our innermost function with respect to... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And then finally, we can take the derivative of our inner function. Let me write it. So we could write this as g prime of h of x. And finally, we just have to take the derivative of our innermost function with respect to x. So the derivative of x squared with respect to x is 2x. So times h prime of x. Let me make every... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And finally, we just have to take the derivative of our innermost function with respect to x. So the derivative of x squared with respect to x is 2x. So times h prime of x. Let me make everything clear. So what we have right over here in purple, this, this, and this are the same things. One expressed concretely, one ex... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
Let me make everything clear. So what we have right over here in purple, this, this, and this are the same things. One expressed concretely, one expressed abstractly. This, this, and this are the same thing, expressed concretely and abstractly. And then finally, this and this are the same thing, expressed concretely an... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
This, this, and this are the same thing, expressed concretely and abstractly. And then finally, this and this are the same thing, expressed concretely and abstractly. But then we're done. All we have to do to be done is to just simplify this. So if we just change the order in which we're multiplying, we have 2x over x ... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
All we have to do to be done is to just simplify this. So if we just change the order in which we're multiplying, we have 2x over x squared. So I can cancel some out. So this 2x over x squared is the same thing as 2 over x. And we're multiplying it times all of this business. So we're left with 2 over x. This goes away... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
So this 2x over x squared is the same thing as 2 over x. And we're multiplying it times all of this business. So we're left with 2 over x. This goes away. 2 over x times the cosine of the natural log of x squared. So it seemed like a very daunting derivative. But we just say, OK, what's the derivative of sine of someth... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
This goes away. 2 over x times the cosine of the natural log of x squared. So it seemed like a very daunting derivative. But we just say, OK, what's the derivative of sine of something with respect to that something? Well, that's cosine of that something. And then we go in one layer. What's the derivative of that somet... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
But we just say, OK, what's the derivative of sine of something with respect to that something? Well, that's cosine of that something. And then we go in one layer. What's the derivative of that something? Well, in that something, we have another composition. So the derivative of ln of x or ln of something with respect ... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
What's the derivative of that something? Well, in that something, we have another composition. So the derivative of ln of x or ln of something with respect to another something, well, that's going to be 1 over the something. So we had gotten a 1 over x squared here. That squared got canceled out. And then finally, the ... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
So we had gotten a 1 over x squared here. That squared got canceled out. And then finally, the derivative of this innermost function. It's like peeling an onion. The derivative of this inner function with respect to x, which was just 2x, which we got right over here. This was 1 over x squared. This was 2x before we did... | Derivative of sin(ln(x_)) Advanced derivatives AP Calculus AB Khan Academy.mp3 |
And what we'll see is it's actually very similar to the definition of any function as the limit approaches infinity. And this is because these sequences really can be just viewed as a function of their indices. So let's say, let me draw an arbitrary sequence right over here. So actually, let me draw it like this, just ... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
So actually, let me draw it like this, just to make it clear what the limit is approaching. So let me draw a sequence that is jumping around a little bit. So let's say when n is equal to 1, a sub 1 is there. When n is equal to 2, a sub 2 is there. When n is equal to 3, a sub 3 is over there. When n is equal to 4, a sub... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
When n is equal to 2, a sub 2 is there. When n is equal to 3, a sub 3 is over there. When n is equal to 4, a sub 4 is over here. When n is equal to 5, a sub 5 is over here. And it looks like as n is, so this is 1, 2, 3, 4, 5. And so it looks like as n gets bigger and bigger and bigger, a sub n seems to be approaching s... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
When n is equal to 5, a sub 5 is over here. And it looks like as n is, so this is 1, 2, 3, 4, 5. And so it looks like as n gets bigger and bigger and bigger, a sub n seems to be approaching some value. It seems to be getting closer and closer. It seems to be converging to some value l right over here. But what we need ... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
It seems to be getting closer and closer. It seems to be converging to some value l right over here. But what we need to do is come up with a definition of what does it really mean to converge to l. So let's say for any, so we're going to say that you converge to l. If for any epsilon greater than 0, for any positive e... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
For any positive epsilon, there is a positive M, capital M, such that if lowercase n is greater than capital M, then the distance between a sub n and our limit, this l right over here, the distance between those two points is less than epsilon. If you can do this for any epsilon, for any epsilon greater than 0, there's... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
So let's parse this. So here I was making the claim that a sub n is approaching this l right over here. I tried to draw it as a horizontal line. This definition of what it means to converge, for a sequence to converge, says look, for any epsilon greater than 0. So let me pick an epsilon greater than 0. So I'm going to ... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
This definition of what it means to converge, for a sequence to converge, says look, for any epsilon greater than 0. So let me pick an epsilon greater than 0. So I'm going to go to l plus epsilon. Actually, let me do it right over here. Let's say this is l plus epsilon, and let's say this is right here. This is l minus... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
Actually, let me do it right over here. Let's say this is l plus epsilon, and let's say this is right here. This is l minus epsilon. So let me draw those two bounds right over here. And so I picked an epsilon here. So for any arbitrary positive epsilon I pick, we can find a positive M. We can find a positive M. So let'... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
So let me draw those two bounds right over here. And so I picked an epsilon here. So for any arbitrary positive epsilon I pick, we can find a positive M. We can find a positive M. So let's say that that is our M right over there. So that as long as our n is greater than our M, then our a sub n is within epsilon of l. S... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
So that as long as our n is greater than our M, then our a sub n is within epsilon of l. So being within epsilon of l is essentially being in this range. This right over here is just saying, look, the distance between a sub n and l is less than epsilon. So that would be any of these. Anything that's in this between l m... | Formal definition for limit of a sequence Series AP Calculus BC Khan Academy.mp3 |
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? Well, we can assume that x for this case ri... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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? Well, 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 th... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 grap... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 it, is just the slope... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 it, is just the slope of the tangent line at any point. So what is the slope of the tangent line at this point... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
Now, remember, the derivative, one way to conceptualize it, 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 horizon... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 I had f of x is equal to 3. Let's say that that's y is equal to 3. What's t... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
The derivative, if I had a function, let's say I had f of x is equal to 3. Let's say that 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? ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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, well, let's use the same a. Let's sa... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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, well, 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 c... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. It's going to be equal to a. I didn't want to do that magenta color. It's going to be e... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
This is going to be equal to a. It's 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. a times the derivative of f of x. Let me do that in blue color. Of f of x. And the other way to denote the derivative of f of x is to just say that this is the... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
Let me do that in blue color. Of f of x. 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 exampl... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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, th... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
Well, this property that I just articulated says, well, this is going to be the same thing as 2 times the derivative. 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 ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 col... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 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 goi... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 10x to the fourth. So 2x to the fifth, you can literally just say, OK, the power rule tells me the derivative of that is 5x to the fourth. 5 times 2 is 10. So that simp... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 of 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 derivativ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. If I were to take the derivative of the sum of two ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
So we 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... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 powe... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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, plus, and over here is negative 4, so it's plus negative 4 times x to the negative 4 minus 1,... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
So it's going to be plus, that's a different shade of blue, plus, 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 can 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... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 sa... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 going to be 2 times 3x squared. What's the derivative of nega... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
Derivative of x to the third is going to be 3x squared, so it's 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 t... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. could say 1x to the 0, but that's just 1. And then finally, what's the derivative of a cons... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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. | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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 ... | Differentiating polynomials Derivative rules AP Calculus AB Khan Academy.mp3 |
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