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Can you find a function whose derivative is 1/x? Created by Sal Khan.
Video transcript
What I wanna do in this video is think about the anti-derivative of one over x, or another way of thinking about it, another way of writing is the anti-derivative x to the negative one power. And we already know if we somehow tried to apply that anti power rule, that inverse power rule over here we would get something that's not defined, we would get x to the zero over zero. Doesn't make any sense. And you might have been saying okay, well I know what to do in this case. When we first learned about derivatives, we know that the derivative, let me just put this in yellow, the derivative with respect to x of the natural log of x is equal to one over x. So why can't we just say that the anti-derivative of this right over here is equal to the natural log of x plus c. And, this isn't necessarily wrong, the problem here is that it's not broad enough. And, when I say it's not broad enough, is that the domain over here, for our original function that we're taking the anti-derivative of, is all real numbers except for x equals 0. So over here x cannot be equal to zero. While the domain over here is only positive numbers. So, over here, x, so for this expression, x has to be greater than zero. So, it would be nice if we could come up with an anti-derivative that has the same domain as the function that we're taking the anti-derivative of. So, it would be nice if we could find an anti-derivative that is defined everywhere that our original function is, so pretty much everywhere except for x equaling zero. So, how can we rearrange this a little bit, so that could be defined for negative values as well? Well one, one possibility is to think about the natural log of the absolute value of X, the natural log of the absolute value of X. So I will put a little question mark here because we don't know really know what the derivative of this thing is going to be. I am not going to rigorously prove it here but I'll, I will give you kind of the conceptual understanding. So to understand it, let's plot, let's plot the natural log of x. And I had done this ahead of time. So that right over there is roughly what the graph of the natural log of x looks like. So what would the natural log of the absolute value of x, is going to look like? Well for positive xs, for positive xs it's gonna look just like this. For positive Xs you take the absolute value of it, It's just the same thing as taking that original value. So it's gonna look just like that for positive xs. But now this is also gonna be defined for negative xs. If you're taking the, the absolute value of negative one, that evaluates to just one, so it's just the natural log of one. So you're gonna be right there. As you get closer and closer and closer to zero from the negative side, you're just gonna take the absolute value. So it's essentially going to be exactly this curve for natural log of x. But the left side of the natural log of the absolute value of x is going to be its mirror image if you were to reflect around the y axis. It's gonna look something like this. It's gonna look something like this. So what's nice about this function is you see it's defined everywhere. It's defined everywhere except for, except for, I'm trying to draw it symmetrically as possible, is defined everywhere except for x equals zero. So, if you combine this pink part and, and this part on the right, if you combine both of these, you combine both of these, you get, you get y is equal to the natural log of the absolute value of x. Now, let's think about its derivative. Well, we already know what the derivative of the natural log of x is. And for positive values of x. So let me write this down. For x is greater than zero, we get the natural log of the absolute value of x is equal to the natural log of x, let me write this. Is equal to the natural log of, is equal to the natural log of x. And we would also know, since these two are equal for x is greater than zero, for x is greater than zero, the derivative. The derivative of the natural log of the absolute value of x, is going to be equal to the derivative, is going to be equal to the derivative of the natural log of x, the natural log of x, which is equal to, which is equal to 1 over x, for x greater than 0. So let's plot that. Let's plot that. I'll do that in gradients, equal to 1 over x. So 1 over x, we've seen it before. It looks something like. It looks something like this. So let me, my best attempt to draw it as both vertical and horizontal isotopes. So it looks something like this, it looks something like this. So this right over here is 1 over x for x is greater than 0. So this is 1 over x when x is greater than 0. So all it's saying here and you can see it pretty clearly, is the slope, the slope right over here. The slope of the tangent line is 1 and so you see that when you look at the derivative slope right over here, the derivative should be equal 1 here. When you get close to zero you have a very, very, steep positive slope here. And so you see you have a very high value for its derivative. And then as you can move away from zero, it's still steep, it's still steep, but becomes less and less and less steep, all the way until you get to one. And then, and then it gets, and then it keeps getting less, and less and less steep. But it never quite gets to an absolutely flat slope. And that's what we see its derivative doing. Now what is the natural log of absolute value of x doing right over here? When we are out here, when we're out here our slope is very close to 0. It's symmetric. The slope here is essentially the negative of the slope here. I could do it maybe clearer showing it right over here. Whatever the slope is right over here, whatever the slope is right over there, it's the exact negative of whatever the slope is at a symmetric point on the other side. So for the other side the slope is right over here, over here it's going to be the negative of that, so it's going to be right, it's going to be right over there. And then the slope just gets more and more and more negative. Right over here the slope. Over here, the slope is a positive one. Over here, it's going to be a negative one. So, right over here, our slope is a negative one. And then as we get closer and closer to zero, it's just gonna get more and more and more negative. So, the derivative of the natural log of the absolute value of x, where x is less than zero, looks something like this. Looks like this. And you see, you see, and it's once again it's not a ultra rigorous proof. But what you see is, is that the derivative of the natural log of the absolute value of x is equal to 1 over x for all xs not equaling 0. So what you're seeing and hopefully you get the, you can visualize that the derivative. Let me write it this way. The derivative, the derivative of the natural log of the absolute value of x is indeed equal to one over x, for, for all, for all x does not equal zero. So this is a much more satisfying anti-derivative for one over x. It has the exact same, has the exact same domain. So when we think about what the anti-derivative is for one over x. And I didn't do a kind of rigorous proof here. I didn't use the definition of the derivative and all of that. But I kinda gave you a visual understanding hopefully of it. We would say it's the natural log of the absolute value of x plus c, and now we have an anti-derivative that has the same domain as the function that we're taking the anti-derivative of.