 Alright, so we're going to be deriving how to get this, which is the formula, I think, for the derivative of the inverse of function. So given a function f of x, what would be the derivative of its inverse? So it'd be 1 over f prime of f of x. f prime is basically just the derivative of f of x. Alright, so the way I was taught this in my calculus class was through implicit differentiation. Basically the x's and y's were flipped, then we used implicit differentiation. Now it wasn't really so much to show how to get that, but it was really more useful for how implicit differentiation worked. Alright, but this is a, but it required a lot of algebra. Well, not that much. It required algebra and it wasn't really all that intuitive. So today I'm going to show you a more intuitive manner that I found, that I came up with. And I always had this in my head and I decided I might as well just make a video about it now. So all you really need to know is how taking in versus work, what slopes, tangent lines and how the relationship to the derivative is, what the relationship to the derivative is. So I'm going to be explaining those really quickly. Let's first start off with taking in inverse. So let's say this is an xy plane and this is our function. When we're taking the inverse of function, basically we're flipping, let's take this point, we're flipping every x to every y. Now, so every y becomes an x and every x becomes a y. Now this isn't a function, but because it has two lines right here. But the basic idea, you can get the basic idea just by looking at how this works. And you're essentially just mirroring it across this line. Alright, so now that we know that, what is a tangent line? So a tangent line is, if we take a point here, we want to get a line that just barely, that only touches that point and not really anything else in the function. I think they say it kisses that point. Of course, of course when you have a function like this and you want to take, okay, you want to take this point, it's going to cross it. But you can get a basic feel for how that works. Like essentially you have a line and you can roll it on that function. That's the way I like to think of it. And the derivative is basically taking the slope of all those tangent lines and mapping them to the x as a function. So if we have x squared, its derivative at every point would be 2x. Alright, so this is how the, alright, so now that we got that down, let's take what would be the inverse of that slope. So let's say we have a line here and this will be a very small line, but I'm making it large for a reason. Because when you have a function, basically you have two little tiny, they say infinitely small, but I don't really like to say infinitely small. It's just really small. We're arbitrarily small. We have these two points. So hand we're doing dy over dx because that's the difference between those two points. And we take the, and that's the slope. Alright, so let's go back to, so let's go back to this slope. If we want to, if we want to know, or if we want to know the slope of this line when it is flipped across or when it's mirrored across this line, we basically do what I just described. So let's define A as x sub A and B similarly as x sub B, y sub B. For taking the slope, we're doing rise over run. So it would be x sub B, wait no, y sub B, because that's rise minus y sub A over x sub B minus y sub minus x sub A. These could be flipped so long as this bottom is also flipped because it would basically be just the negative of this and also the negative of this top and negative one times negative over negative one equals one. Alright, so when we're taking this inverse, again just flipping flipping x with y and y with x. So it would look like this. x sub B minus x sub A, that's terribly over y sub A minus y sub B minus y sub A. And that is just the inverse, the inverse of those points. So it just be, it's just this flipped around. So like one over x, one over that over over the slope. Okay. So now that we have, so now that we know what this is, let's hit one over s, we can map every tangent line with this inverse function. So again, let, let's have this PR inverse function, this PR original function. So what we're doing here is we're going to want to go backwards. So let's say we want to take the tangent line at this point right here. And let's define this point as x and f on the value of a negative one x. That looks more like a four than a negative one. Alright. And this is how, this is how this point is defined. What we want to find is that same point, but on the original function. So basically we want, I don't know how to explain this, but we want to flip it again. We want to flip it again through this line because we can, we can get the, we can get the derivative of this function. Okay. So when we flip it, we're basically taking, we're basically doing this. Again, we're flipping these two like that. So let's do that and let's just say that it's this. Okay. Now we just take the slope, which is f prime and we're sticking negative out negative. I mean the inverse of the original function into it. Alright. And now we have this slope right here, but that's not the same as this slope right here. We need to, since because we're flipping it across this line, then the inverse or whatever, we're getting one over that slope. So this is how we get one over f prime of the inverse function. So yeah, that's, that's pretty much it. I feel like it's better than deriving it through implicit differentiation.