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Current time:0:00Total duration:4:44

Addressing treating differentials algebraically

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.D (LO)
,
FUN‑7.D.1 (EK)
,
FUN‑7.D.2 (EK)

Video transcript

so when you first learn calculus you learned that the derivative of some function f could be written as f prime of x is equal to the limit as and there's multiple ways of doing this the change in X approaches zero of f of X plus our change in X minus f of X over our change in X and you learn multiple notations for this for example if you know that Y is equal to f of X you might write this as Y prime you might write this as dy/dx which you'll often hear me say is the derivative of Y with respect to X and that you could view the derivative of F with respect to X because Y is equal to our function but then later on when you especially when you start getting into differential equations you see people start to treat this notation as an actual algebraic expression for example you will learn or you might have already seen if you're trying to solve the differential equation the derivative of Y with respect to X is equal to Y so the rate of change of Y with respect to X is equal to the value of y itself this is one of the the most basic differential equations you might see you'll see this technique where people say well let's just multiply both sides by DX just treating DX like as if it's some algebraic expression so you multiply both sides by DX and then you have so that would cancel out algebraically and so you see people treat it like that so you have dy is equal to Y times DX and then they'll say okay let's divide both sides by Y which is a reasonable thing to do y is an algebraic expression so if you divide both sides by Y you get 1 over Y DUI is equal to DX and then folks will integrate both sides to find a general solution to this differential equation but my point on this video isn't to think about how do you solve a differential equation here but to think about this notion of using what we call differentials so a DX or dy and treating them algebraically like this treating them as algebraic expressions where I can just multiply both sides by just DX or dy or divide both sides by DX or dy and I don't normally say this but they're the the rigor you need to show that this is okay in this situation is not an easy thing to say and so to just feel reasonably okay about doing this this is a little bit hand wavy it's not super mathematically rigorous but it has proven to be a useful tool for us to find these solutions and conceptually the way that I think about a dy or a DX is this is the super small change in Y in response to a super small change in X and that's essentially what's what this definition of the limit is telling us especially as Delta X approaches zero we're going to have a super small change in X as Delta X approaches zero and then we're going to have a resulting super small change in Y so that's one way that you can feel a little bit better of and this is actually one of the justifications for this type of notation is you could view this what's the resulting super small or what's the super small change in Y for a given super small change in X which is giving us the sense of what's the limiting value of the slope as we go from the slope of a secant line to a tangent line and if you view it that way you might feel a little bit better about using using the differentials or treating them algebraically whereas okay let me just multiply both sides by that super small change in X so the big picture is this is a technique that you will often see in in introductory differential equations classes introductory multivariable classes and introductory calculus classes but it's not very mathematically rigorous to just treat differentials like algebraic expressions but even though it's not very mathematically rigorous to do it willy-nilly like that it has proven to be very useful now as you get more sophisticated in your mathematics there are rigorous definitions of a differential where you can get a better sense of where it is mathematically rigorous to use it and where it isn't but the whole point here is if you felt a little weird feeling about multiplying both sides by DX or dividing both sides by DX or dy you your feeling was is mathematically justified because it's not a very rigged first thing to do at least until you have more rigor behind it but I will tell you that if you're an introductory student it is a reasonable thing to do as you explore and manipulate some of these basic differential equations