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Now I introduce you to the concept of exact equations.

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And it's just another method for solving a certain type of

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differential equations.

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Let me write that down.

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Exact equations.

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Before I show you what an exact equation is, I'm just

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going to give you a little bit of the building blocks, just

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so that when I later prove it, or at least give you the

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intuition behind it, it doesn't seem like it's coming

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out of the blue.

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So let's say I had some function of x and y, and we'll

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call it psi, just because that's what people tend to use

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for these exact equations.

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So psi is a function of x and y.

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So you're probably not familiar with taking the chain

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rule onto partial derivatives, but I'll show it to you now,

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and I'll give you a little intuition, although

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I won't prove it.

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So if I were to take the derivative of this with

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respect to x, where y is also function of x, I could also

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write this as y-- sorry, it's not y, psi.

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Undo.

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So I could also write this as psi, as x and y, which is a

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function of x.

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I could write it just like that.

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These are just two different ways of

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writing the same thing.

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Now, if I were to take the derivative of psi with respect

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to x-- and these are just the building blocks-- if I were to

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take the derivative of psi with respect to x, it is equal

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to-- this is the chain rule using partial derivatives.

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And I won't prove it, but I'll give you the

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intuition right here.

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So this is going to be equal to the partial derivative of

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psi with respect to x plus the partial derivative of psi with

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respect to y times dy dx.

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And this is should make a little bit of intuition.

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I'm kind of taking the derivative with respect to x,

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and if you could say, and I know you can't, because this

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partial with respect to y, and the dy, they're

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two different things.

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But if these canceled out, then you'd kind of have

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another partial with respect to x.

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And if you were to kind of add them up, then you would get

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the full derivative with respect to x.

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That's not even the intuition, that's just to show you that

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even this should make a little bit of intuitive sense.

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Now the intuition here, let's just say psi, and psi doesn't

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always have to take this form, but you could use this same

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methodology to take psi to more complex notations.

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But let's say that psi, and I won't write that it's a

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function of x and y.

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We know that it's a function of x and y.

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Let's say it's equal to some function of x, we'll call that

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f1 of x, times some function of y.

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And let's say there's a bunch of terms like this.

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So there's n terms like this, plus all the way to the nth

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term is the nth function of x times the nth function of y.

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I just defined psi like this just so I can give you the

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intuition that when I use implicit differentiation on

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this, when I take the derivative of this with

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respect to x, I actually get something that

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looks just like that.

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So what's the derivative of psi with respect to x?

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And this is just the implicit differentiation that you

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learned, or that you hopefully learned, in your first

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semester calculus course.

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That's equal, and we just do the product rule, right?

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So the first expression, you take the derivative of that

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with respect to x.

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Well, that's just going to be f1 prime of x times the second

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function, well, that's just g1 of y.

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Now you add that to the derivative of the second

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function times the first function.

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So plus f1 of x, that's just the first function, times the

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derivative of the second function.

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Now the derivative of the second function, it's going to

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be this function with respect to y.

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So you could write that as g1 prime of y.

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But of course, we're doing the chain rule.

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So it's that times dy dx.

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And you might want to review the implicit differentiation

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videos if that seems a little bit foreign.

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But this right here, what I just did, this expression

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right here, this is the derivative with

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respect to x of this.

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And we have n terms like that.

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So if we keep adding them, I'll do them vertically down.

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So plus, and then you have a bunch of them, and the last

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one's going to look the same, it's just the

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nth function of x.

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So fn prime of x times the second function, g n of y,

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plus the first function, fn of x, times the derivative of the

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second function.

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The derivative of the second function with respect to y is

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just g prime of y times dy dx.

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It's just a chain rule.

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dy dx.

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Now, we have two n terms. We have n terms here, right,

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where each term was a f of x times a g of y, or f1 of x

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times g1 of y, and then all the way to fn of

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x times gn of y.

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Now for each of those, we got two of them when we did the

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product rule.

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If we group the terms, so if we group all the terms that

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don't have a dy dx on them, what do we get?

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If we add all of these, I guess you could call them on

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the left hand side, I'm just rearranging, it all equals f1

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prime of x times g1 of y, plus f2, g2, all the way to fn

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prime, I'm sorry, fn prime of x, gn of y.

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That's just all of these added up, plus all

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of these added up.

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All the terms that have the dy dx in them.

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And I'll do them in a different color.

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So all of these terms are going to be

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in a different color.

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I'll do it in a different parentheses.

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Plus f1 of x g1 prime of y, and I'll do the dy dx later,

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I'll distribute it out.

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Plus, and we have n terms, plus fn of x gn prime of y,

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and then all of these terms are multiplied by dy dx.

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Now, something looks interesting here.

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We originally defined our psi, up here, as this right here,

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but what is this green term?

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Well, what we did is we took all of these individual terms,

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and these green terms here are just taking the derivative

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with respect to just x on each of these terms. Because if you

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take the derivative just with respect to x of this, then the

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function of y is just a constant, right?

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If you were to take just a partial derivative with

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respect to x.

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So if you took the partial derivative with respect to x

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of this term, you treat a function of y as a constant.

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So the derivative of this would just be f prime of x, g1

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of y, because g1 of y is just a constant.

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And so forth and so on.

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All of these green terms you can view as a partial

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derivative of psi with respect to x.

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We just pretended like y is a constant.

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And that same logic, if you ignore this, if you just look

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at this part right here, what is this?

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We took psi, up here, we treated the functions of x as

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a constant, and we just took the partial derivative with

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respect to y.

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And that's why the primes are on all the g's.

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And then we multiply that times dy dx.

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So you could write this, this is equal to-- I'll do this

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green-- this green is the same thing as the partial of psi

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with respect to x.

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Plus, what's this purple, this part of the purple?

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Let me do it in a different color, in magenta.

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This, right here, is the partial of psi with respect to

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y, and then times dy dx.

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So that's essentially all I wanted to show you right now

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in this video, because I realize I'm almost

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running out of time.

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That the chain rule, with respect to one of the

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variables, but the second variable in the function is

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also a function of x, the chain rule is this.

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If psi is a function of x and y, and I would take not a

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partial derivative, I would take the full derivative of

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psi with respect to x, it's equal to the partial of psi

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with respect to x, plus the partial of psi with respect to

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y, times dy dx.

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If y wasn't a function of x, or if y if it was independent

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of x, than dy dx would be 0.

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And this term would be 0, and then the derivative of psi

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with respect to x would be just the partial of psi with

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respect to x.

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But anyway, I want you to just keep this in mind.

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And in this video I didn't prove it, but I hopefully gave

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you a little intuition if I didn't confuse you.

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And we're going to use this property in the next series of

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videos to understand exact equations a little bit more.

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I realize that in this video I just got as far as kind of

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giving you an intuition here.

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I haven't told you yet what an exact equation is.

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I will see you in the next video.

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