Implicit Differentiation

Normally, I don’t have trouble teaching implicit differentiation. However, I’m never satisfied with what I do. I’m fairly certain that I have taught it four different ways in the past four years. But what’s common is that we do a lot of algebra. By the end, they can find \frac{dy}{dx} for a relation like \sin(xy)+y^3=2x+y. Or something like that. But we lose the meaning of what we’re doing.

I realized we can do all this algebra, but it’s all procedure. And so there’s no real depth.

So today, after introducing implicit differentiation (including some visual motivation), I assigned 5 basic problems from the textbook. Each of the problems has an equation like 3y^3+x^2=5 and students are asked to find \frac{dy}{dx}. My kids are going to go home today and struggle with it. We’ll spend about 20 or 25 minutes in our next class going over their solutions, talking about things, whatever.

And then… then… I’m going to hand out this sheet I wrote today.

[.doc, .pdf]
[if you’re wondering, the graphs were made by the fabulous winplot which I adore… it can do implicit plotting!]

My kids found \frac{dy}{dx} for homework. Now in class, my kids are going to interrogate what that means.

I am not sure yet how I’m going to structure the class. I think I might have us all work together on the first problem (#9), and then assign pairs to work on two of the remaining problems. And then I’ll pick one problem for each pair to present to the class. But what I’m truly happy about is that each problem gets kids to relate implicit differentiation to a graphical understanding of the derivative. It forces my kids to look at the derivative equation, and make connections to the original graph.

Although I’m proud of it, I’m honestly just not sure if this investigation is beyond the scope of my kids’s abilities. It pulls together a lot of concepts. I think it’ll work for them. This year I have a really really strong crew so I have faith. However, it’s an activity I’m going to have to give my kids time to do, and room to struggle. I know me, and I’m going to want to rush it, and I’m going to want to help them in ways that aren’t good for them. The struggle is where they’re going to learn in this, so I have to give it time and stay out.

I am in the middle of a hellish week, but if I have time, I’ll try to report back how it goes after we do it in class.



  1. Cool stuff. I also recommend using at least one example that they “already know”, maybe something like x^2 + y^2 = 25 and/or x^2 – y^2 = 25. I always liked relating implicit back to students’ existing knowledge of the velocity and acceleration in circular motion.

    – Bowen

  2. Hihi Bowen,

    Yes! Yes! I actually started out with existing knowledge yesterday. We worked with a simple rational function one written strangely… yx + y +1 = x… first by having them “take what they don’t know and turn it into what they do know” (which means, since we hadn’t yet gone over implicit diff, rewriting it as a simple rational function)… and then going over that same problem with implicit diff and getting a different answer (one involving y!).

    and we worked with x^2+y^2=25 also, and saw it work out graphically. This worksheet is about taking it a bit further.


  3. I hate how we’re forced to teach this in the calculus courses here at (generic big midwest state university). There’s basically one-and-a-half class-periods for implicit differentiation (one lecture, one recitation but the recitation also covers logarithmic differentiation) and that’s it; but that’s not the bad part. The bad part is that the course is centrally coordinated and there’s no way of knowing whether or not the tests are going to have problems involving “hard” algebra (e.g., dy/dx is inside of parentheses and you must expand before solving). So, we’re more or less forced to assume the worst and teach exactly that, which makes any sort of conceptual work pretty much impossible given the time constraints. (One could argue that such algebra shouldn’t be “hard” at all considering the students are theoretically supposed to already be competent at algebra, but if you want to argue that, you’re in for a miserable quarter…)

  4. Haha you don’t think I’d argue the last point! I’m a high school teacher. It’s still a very common occurrence to have seniors simplify \frac{x+2}{x} to be 2. Because, you know, you cancel out the 2s.

    It may be my fault, seeing as I taught a lot of them Algebra II.

    Yeah, just blame me. ;)

  5. As a student and hobbiest mathematician, I don’t entirely see the problem with the algebra you mention in your normal method of teaching implicit differentiation. The concept of implicit differentiation is that we can take the derivitive with respect to anything. Once we do that, we have (hopefully) reduced the problem to something we can solve. If you are concerned about loosing meaning, something that might be effective is saying ‘now we can use algebra to simplify this and arrive at…’, drawing attention to the fact that the algebra is now a tool, not the problem.

  6. I do still teach implicit differentiation in the “normal method.” The problem is that I used to teach it, and then move on. And I never felt like the students could draw a connection between the algebra, and the meaning of the algebra. The fact that a y appeared in the derivative is weird. And the fact that you can do a lot more conceptual work connecting the equation of the derivative to the graph of the relation, well, as a teacher that’s powerful stuff. It reinforces what they already know and stretches their minds with the new sorts of questions I can pose (like I did on the worksheet).

    So I do teach it normally, but then I have them go a little bit deeper.

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