Exact equations intuition 1 (proofy)

now introduce you to the concept of exact equations and it's just another method for solving a certain type of differential equations let me write that down exact exact equations equations before I show you an exact equation is I'm just going to give you a little bit of the building blocks just so that when I later prove it or at least give you the intuition behind it you it doesn't you know seem like it's coming out of the blue so let's say I had some function of x and y and we'll call it sy just because that's what people tend to use for these exact equations so as AI is a function of x and y is AI is a function of x and y so you're probably not familiar with taking taking the chain rule on two partial derivatives but I'll show it to you now and I'll give you a little intuition although 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 I could uh maybe I could also write this as Y sorry not Y sy undo so I could also write this as I as x + 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 Z with respect to X and these are just the building blocks if I were to take the derivative of Y with respect to X it is equal to it this is the chain rule using partial derivatives and I won't prove it but I'll give you the intuition right here so this is going to be equal to the partial derivative of Z with respect to X plus the partial derivative of Z with respect to Y times dy DX times dy DX and this should make a little bit of intuition right I'm thinking you know I'm kind of taking the derivative with respect to X and then I'm and then if 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 these cancelled out then you'd kind of have another partial with respect to X and when you were to if you were to kind of add them up then you would get you know the full derivative with back to X that's not even in the intuition that's just to kind of show you that even this should make a little bit of intuitive sense now the intuition here let's just say and I'm not let's just say zai and it's I doesn't always have to take this form but you could you could use the same methodology to take is either kind of more complex notations but let's say that zai 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 it's equal to f some function of x call that f 1 of x times some function of Y now let's 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 define design like this it's just well I can give you the intuition that when I use implicit differentiation on this when I take the derivative of this with respect to X I actually get something that looks just like that so what's the derivative of Z with respect to X the derivative of Z 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 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 well that's just going to be f1 prime of x times the second function well that's just G 1 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 derivative the second function it's going to be this function with respect to Y so you can write that is g1 prime of Y but of course we're 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 this is the derivative with respect to X of this right and 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 ones going to look the same it's just it's the Entune of X so f n 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 the second function with respect to Y is just G prime of Y times dy DX that's just the chain rule dy/dx now we have well now we have two n terms we had n terms here right where each term was a f of X times the G of Y or F 1 of x times G 1 of Y and then all the way to FN of x times GN of Y now we have 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 the terms that don't have a dy/dx on them what do we get if we add all of these I guess you could call them on the left-hand side you get I'm just rearranging it all equals f1 prime of x times G 1 of Y plus F 2 G 2 all the way to F n Prime sorry F n prime of X G n of Y that's just all of these added up plus plus all of these added up all the terms all of the terms that have the dy/dx in them right so those are I'll do them in a different color let me so all of these terms are going to be the different color I'll go to different parenthesis plus F 1 of X G 1 prime of Y and I'll do the dy/dx later I'll distribute it out plus and we have n terms plus FN of X gee and prime of Y and then all of these terms are multiplied by dy/dx now something looks interesting here right we originally defined hours I appear as as this right here but what is this green term well what we did is we took all of these individual terms and this green these green terms here are just taking the derivative with respect to just X on each of these terms right because if you take the derivative 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 with respect 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 G 1 of Y right because G 1 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 Z with respect to X we just we just pretended like Y is a constant and that same logic if you ignore this if you just look at this part right here what is this we took sy up here we treated the functions of X as a constant we treated the functions 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 is equal to I'll do this green this could be this Green is the same thing as the partial of Z with respect to X plus what's this purple that this part of the purple let me do a different color in a magenta this right here is the partial of Z with respect to Y and then times dy DX so that's essentially all I wanted to show you right now in this video because I realize I'm almost running of time that when you the chain rule when you're taking with respect to one of the variables but both of the vert but you know the function the second variable of the function is also a function of X the chain rule is this if f of Z is a function of x and y and I take not a partial derivative I take the full derivative of Z with respect to X is equal to the partial of Z with respect to X plus the partial of Z with respect to Y times dy DX if why wasn't the function of X or if Y in no way was it was independent of X then dy DX would be 0 and this term would be 0 and then the derivative of Z with respect to X would be just the partial of Z with respect to X but anyway I want you to I want you to just keep this in mind and I'd in this video I didn't prove it but I hopefully gave you a little intuition if I didn't confuse you and we're going to use this property in the next series of videos to understand exact equations a little bit more I realize in this video I just got as far as it's 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