Addressing treating differentials algebraically | AP Calculus AB | Khan Academy

So when you first learn calculus, you learn that the derivative of some function f could be written as f prime of x is equal to the limit as the change in x approaches zero of f of x plus the change in x minus f of x over the change in x.

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. You could use the derivative of f with respect to x because y is equal to our function.

But then later on, 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 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 it's some algebraic expression. So you multiply both sides by dx, and then you have dy is equal to y * dx. 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/y dy is equal to dx. Then folks will integrate both sides to find a general solution to this differential equation. But my point in this video isn't to think about how to solve a differential equation, 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.

I don't normally say this, but there is the rigor you need to show that this is okay in this situation. It is not an easy thing to say. 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.

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. That’s essentially 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.

That’s one way that you can feel a little bit better about this. This is actually one of the justifications for this type of notation: you could view this as what's the resulting 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. If you view it that way, you might feel a little bit better about using the differentials or treating them algebraically.

So the big picture is this is a technique that you will often see 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. 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 about multiplying both sides by dx or dividing both sides by dx or dy, your feeling was mathematically justified because it's not a very rigorous thing to do, at least until you have more rigor behind it.

But I will tell you, if you're an introductory student, it is a reasonable thing to do as you explore and manipulate some of these basic differential equations.