Now introduced you to the concept of exact equation. And it’s just another method for solving a certain type of differential equation, let me write that down. Exact equations. Before I show you an exact equation is, I'm just give you a little bit of the building blocks, it’s just so that when I later prove it or at least give you the intuition behind it, it doesn’t seem like its coming out of the blue.
So let's say I had some function of x and y and we’ll call it psi. Just because that’s what people tend to use with this exact equations. So psi is a function of x and y. so you're probably not familiar with taking the chain rule on to partial derivatives, but I’ll show it to you now. I’ll give you a little intuition, though I won't prove it.
So if I were to take the derivative of this with respect to x, where y is also a function of x. maybe I could also write this as psi as x and y which is a function of x. I could write it just like that. These are just two different ways of writing the same thing. Now if I were to take the derivative of psi with respect to x, and these are just the building blocks. If I were to take the derivative of psi with respect to x, it is equal to, this is the chain rule using partial derivatives. And I won't prove it, but I will give you the intuition right here. So this is going to be equal to the partial derivative of psi with respect to x plus the partial derivative of psi with respect to y times dy/dx. And this should make a little bit of intuition. I'm taking the derivative with respect to x and then if you could say, and I know you can't, because this partial with respect to y and the dy. They're two different things, but if this cancelled out, then you kind of have another partial with respect to x. and when you were to kind of add them up, then you would get the full derivative with respect to x. that’s not even the intuition, that’s to kind of show you that even this should make a little bit intuitive sense.
Now the intuition here, let's just say psi. And psi doesn’t always have to take this form, but you could use the same methodology to take psi to a more complex notations. But let's say that psi, and I won't write this, it’s a function of x and y. we know it’s a function of x and y. let's say its equal to some function of x, we call that f1 of x. times some function of y. and say there's a bunch of terms like this. So there's n terms like this. Plus all the way in the nth term is the nth function of x times the nth function of y. I just defined the psi like this just so I can give you the intuition that when I used implicit differentiation on this, when I take the derivative of this with respect to x. I actually get something that looks like that.
So what's the derivative of psi with respect to x? The derivative of psi with respect to x, and this is just the implicit differentiation that you learned in your first or that you hopefully learned in your first semester of calculus course. That’s equal, we just do the product rule, right? So the first expression, you take the derivative of that with respect to x. that’s just going to be f1 prime of x times the second function. Well that’s just g1 of y. now you add that to the derivative of the second function times the first function. So plus f1 of x, that’s just the first function times the derivative of the second function. Now the derivatives of the second function, it's going to be this with respect to y. so you can write that as g1 prime of y. well of course were doing the chain rule. So that times dy/dx. And you might want to review the implicit differentiation videos if this seems a little bit foreign. But this right here, what I just did, this expression right here is the derivative with respect to x of this, right? Now we have n terms like that, so if we keep adding them, I’ll do them vertically down. So plus and then you have a bunch of them. And then the last one will going to look the same. It’s the nth function of x. so fn prime of x times the second function gn of y plus the first function, fn of x, times the derivative of the second function. The derivative of second function with respect to y is just g prime of y times dy/dx. it’s just the chain rule.
Now we have two n term. We have n terms here, right? where each term is f of x times g of y or f1 of x times g1 of y and then all the way to fn of x times gn of y. now for each of those, we got two of them when we did the product rule. If we group the terms, so if we group all of the terms and we don’t have a dy/dx on them, what do we get? If we add all of these, I guess we can call them on the left hand side, it all equals f1 prime of x times g1 of y plus f2g2 all the way to fn prime of x gn of y. that’s just all of these added up. Plus all of these added up, all of the terms that have the dy/dx in them. Right? so those are, I’ll do them in different color. so all of these terms, plus f1 of x g1 prime of y. and I’ll do the dy/dx later, I’ll just distribute it out. Plus and we have n terms, fn of x gn prime of y. and then all of these terms are multiplied by dy/dx.
Now, something looks interesting here. Right? We originally defined our psi up here as this right here. but what is this green term? Well, what we did is we took all of these individual terms and these green terms are just taking the derivative with respect to just x on each of these terms. Right? but when you take the derivative of just with respect to x of this, then the function of y is just a constant. Right? If you were to take just the partial derivative with respect to x. So if you take the partial derivative to x of this term, you treat a function of y as a constant. so the derivative of this would just be f prime of x g1 of y. Right? Because g1 of y is just a constant. And so forth and so on. All of these green terms, you can view as the partial derivative of psi with respect to x. We just pretended like y is a constant. And that same logic, if you just look at this part right here, what is this? We took psi up here, we treated the function of x as a constant and we just took the partial derivative with respect to y. And that’s why the primes are on all the g’s. And then we multiply that times dy/dx. so you could write this, this green is the same thing as the partial of psi with respect to x plus, what's this purple? This part of the purple. Let me do it in different color, let me do it magenta. This right here is the partial of psi with respect to y and then times dy/dx.
So that’s essentially what I wanted to show you right now on this video. Because I realized I'm almost running out of time. the chain rule, with respect to one of the variables but the second variable in the function is also a function of x, the chain rule is this. If psi is a function of x and y, and I’ll take not the partial derivative, if I take the full derivative of psi with respect to x is equal to the partial of psi with respect to x plus the partial of psi with respect to y times dy/dx. If y wasn’t a function of x or if y is independent of x, then dy/dx would be 0. And this term would be 0 and then the derivative of psi with respect to x would be just the partial of psi with respect to x.
But anyway, I want you to just keep this in mind. And in this video, I didn’t prove it, but I hopefully gave you a little intuition if I didn’t confuse you. And were going to use this property in the next series of videos to understand exact equations a little bit more. I realized in this video I just got as far as kind of giving you an intuition here. I haven’t told you yet what an exact equation is. I will see you in the next video.
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