So this kid has clearly seen some examples worked out on the board and noticed that something weird happens with implicit differentiation, i.e., we sometimes get these extra factors of dy/dx that don’t show up in quite the same way as when y is an explicit function of x. Hence the first line? When in doubt, use a shotgun.

The second line looks better. Inexplicably better. My first question would be, “What happened to all these (dy/dx)es between line 1 and line 2?” That might shed some serious light on things for me. Starting with the second line, though, we should focus on the product rule, the application of which seems to be wildly inconsistent.

I think we should start by doing some explicit differentiation with product rule, d/dx [f(x)g(x)], and then replace one of those functions with y to emphasize that that’s where the dy/dx terms come from; they’re placeholders for the unknown derivative of y with respect to x. Finally, work in some more complicated expressions involving y so we can practice both the implicit differentiation piece and the product rule piece simultaneously—the combination of which is responsible for at least part of the wackiness going on here.

On the plus side, the algebraic manipulation seems solid, which is often a nontrivial element of these imp diff problems; lots of students can blindly apply term-by-term differentiation rules of thumb (i.e. “do the calculus part”) but still really struggle with basic algebra/function concepts.

I’m no Calculus pro, but my thought was that the student was using dy/dx on that first line as a differential operator.

Ooh, I didn’t think of that. All those dy’s really threw me; if we’re going to through the operator in front of ever term, it should be d/dx (the derivative of whatever this thing, not necessarily y, is with respect to x). Maybe I wasn’t giving the kid enough credit here. If that’s the case, then it explains, for instance, the correct differentiation of the first term. Also where a bunch of those things disappeared to in the second line.

I suppose you just shed the light I was hoping to have shed by the student with my first question from above. And I think it goes without saying that I’m no calculus pro, either.

My thoughts on the product rule, I think, stand.

Ya, I also think they mis-wrote dy/dx for d/dx in the first line. Seems quite possible that they don’t really know what dy/dx and d/dx mean and are memorizing steps.

Not using the product rule which is a common mistake. The majority of students will recognize this mistake if you point to the spot and ask them what they forgot to do. The overwhelming majority of students have no idea why the product rule works.

The more serious misunderstanding is that taking the derivative of a terms like y^3 (with resprect to x) involves the chain rule. For terms that only have y’s, they are just re-writing the term and pasting a dy/dx at the end.

I think you have to go back to the idea that, if y was ln(x) + 5, then y^3 would just be (ln(x) + 5)^3, and we would use the chain rule to differentiate that term.

Am I the only one that doesn’t like the dy/dx notation?

It is especially likely that the student does not see the difference between d/dx and dy/dx given the use of the prime notation everywhere else in this problem. This is something I struggle with each year, do I present multiple notations early on since they will see them, or do I stick with the d/dx operator notation since it gets to the heart of the relationship to our old slope formulas? This year, my 20th of teaching Calculus, I think I’ll try to stick to d/dx more regularly

Yeah, I’m glad you guys both caught the operator notation/notation error. Also, your point about the chain rule is more important than my point about the product rule. I’m going to go back to commenting on algebra mistakes from now on, because I’m apparently lousy at reading calculus student minds.

And you’re not the only one who dislikes dy/dx. Euler, Lagrange, and Newton—at a minimum—didn’t like it, so I reckon you’re in good company.

I was taken aback by all of the dy/dx’s in the first step too, but then I realized “what the student meant.” (How often do we have to do that? 🙂 )

Initially just the neglected product rule jumped out, but then I also noticed the lack of chain rule application. Maybe backing up a bit and differentiating something like y^3 = (a bunch of stuff with only x) FIRST by solving for y, so you have to apply the chain rule to all of the “garbage inside the parentheses.” THEN try it using implicit differentiation. Will the results “look” the same? No, unless you take the second one, solve it for y and plug it in. Hopefully it will serve as a reminder that, in reality, y “represents” some expression involving x, and you are differentiating with respect to x!

I agree that this student has DEFINITELY seen some worked examples without really internalizing/understanding “why.”

I’m going to try to respond without reading everyone else’s analyses…

First off, on the first line it’s pretty clear that the student is trying to say “d/dx” but doesn’t understand that “dy/dx” is not the same thing. S/he is confused about that notation, for sure. (It’s a SUPER common issue I’ve seen.)

And then, of course, the second line is a mess, but in a very predictable way. The RHS was differentiated correctly and the first term on the LHS was differentiated correctly. Those are standard things that the student has certainly seen before.

However, the second and third terms on the LHS indicate the student is treating “y” like it’s a constant (a less common mistake), and has incorrectly devised the rule “whenever you see a y in a term, stick on a y prime” (a more common mistake).

The main misconception is that the student is not interpreting y as a variable, and more importantly, a variable that is dependent on x. (And thus, because of that, the product and chain rules both come into play.)

Although I have only been teaching calculus for 5 years, I have found that it has gotten easier to teach kids the algebraic steps to do implicit differentiation, and they eventually get it. But it has been a much harder uphill battle to get them to understand (a) what exactly the meaning of what they are doing is and (b) what the connection is to everything else is. I’m getting better at it, but this throws so many abstract concepts at them that I can understand why it’s challenging.

Implicit differentiation, when I saw it first at school, left me feeling really uneasy. We had been shown differentiation of real functions and given some proofs about how that worked and what it meant. Then, without a warning, we see differentiation applied to something that is not a function. Huh?

Now, many years later, I know this was wrong. At the very least we should have been warned that something new was happening and told that a justification did exist, but was complicated, so for now all we had to do was learn the process.

My eldest son has recently been learning calculus and the teacher did exactly the same to him as was done to me, decades earlier.

It’s no wonder that students do not understand implicit differentiation well and just try to copy what they have seen, rather inaccurately.

The notation for calculus and the style of writing implicit differentiation could be improved, and so could the teaching of it. Why this has not happened in over 30 years I do not know.

## 9 replies on “Implicit Differentiation”

So this kid has clearly seen some examples worked out on the board and noticed that something weird happens with implicit differentiation, i.e., we sometimes get these extra factors of dy/dx that don’t show up in quite the same way as when y is an explicit function of x. Hence the first line? When in doubt, use a shotgun.

The second line looks better. Inexplicably better. My first question would be, “What happened to all these (dy/dx)es between line 1 and line 2?” That might shed some serious light on things for me. Starting with the second line, though, we should focus on the product rule, the application of which seems to be wildly inconsistent.

I think we should start by doing some explicit differentiation with product rule, d/dx [f(x)g(x)], and then replace one of those functions with y to emphasize that

that’swhere the dy/dx terms come from; they’re placeholders for the unknown derivative of y with respect to x. Finally, work in some more complicated expressions involving y so we can practice both the implicit differentiation piece and the product rule piece simultaneously—the combination of which is responsible for at leastpartof the wackiness going on here.On the plus side, the algebraic manipulation seems solid, which is often a nontrivial element of these imp diff problems; lots of students can blindly apply term-by-term differentiation rules of thumb (i.e. “do the calculus part”) but still really struggle with basic algebra/function concepts.

I’m no Calculus pro, but my thought was that the student was using dy/dx on that first line as a differential operator.

Ooh, I didn’t think of that. All those dy’s really threw me; if we’re going to through the operator in front of ever term, it should be d/dx (the derivative of whatever this thing, not necessarily y, is with respect to x). Maybe I wasn’t giving the kid enough credit here. If that’s the case, then it explains, for instance, the correct differentiation of the first term. Also where a bunch of those things disappeared to in the second line.

I suppose you just shed the light I was hoping to have shed by the student with my first question from above. And I think it goes without saying that I’m no calculus pro, either.

My thoughts on the product rule, I think, stand.

Ya, I also think they mis-wrote dy/dx for d/dx in the first line. Seems quite possible that they don’t really know what dy/dx and d/dx mean and are memorizing steps.

Not using the product rule which is a common mistake. The majority of students will recognize this mistake if you point to the spot and ask them what they forgot to do. The overwhelming majority of students have no idea why the product rule works.

The more serious misunderstanding is that taking the derivative of a terms like y^3 (with resprect to x) involves the chain rule. For terms that only have y’s, they are just re-writing the term and pasting a dy/dx at the end.

I think you have to go back to the idea that, if y was ln(x) + 5, then y^3 would just be (ln(x) + 5)^3, and we would use the chain rule to differentiate that term.

Am I the only one that doesn’t like the dy/dx notation?

It is especially likely that the student does not see the difference between d/dx and dy/dx given the use of the prime notation everywhere else in this problem. This is something I struggle with each year, do I present multiple notations early on since they will see them, or do I stick with the d/dx operator notation since it gets to the heart of the relationship to our old slope formulas? This year, my 20th of teaching Calculus, I think I’ll try to stick to d/dx more regularly

Yeah, I’m glad you guys both caught the operator notation/notation error. Also, your point about the chain rule is more important than my point about the product rule. I’m going to go back to commenting on algebra mistakes from now on, because I’m apparently lousy at reading calculus student minds.

And you’re not the only one who dislikes dy/dx. Euler, Lagrange, and Newton—at a minimum—didn’t like it, so I reckon you’re in good company.

I was taken aback by all of the dy/dx’s in the first step too, but then I realized “what the student meant.” (How often do we have to do that? 🙂 )

Initially just the neglected product rule jumped out, but then I also noticed the lack of chain rule application. Maybe backing up a bit and differentiating something like y^3 = (a bunch of stuff with only x) FIRST by solving for y, so you have to apply the chain rule to all of the “garbage inside the parentheses.” THEN try it using implicit differentiation. Will the results “look” the same? No, unless you take the second one, solve it for y and plug it in. Hopefully it will serve as a reminder that, in reality, y “represents” some expression involving x, and you are differentiating with respect to x!

I agree that this student has DEFINITELY seen some worked examples without really internalizing/understanding “why.”

I’m going to try to respond without reading everyone else’s analyses…

First off, on the first line it’s pretty clear that the student is trying to say “d/dx” but doesn’t understand that “dy/dx” is not the same thing. S/he is confused about that notation, for sure. (It’s a SUPER common issue I’ve seen.)

And then, of course, the second line is a mess, but in a very predictable way. The RHS was differentiated correctly and the first term on the LHS was differentiated correctly. Those are standard things that the student has certainly seen before.

However, the second and third terms on the LHS indicate the student is treating “y” like it’s a constant (a less common mistake), and has incorrectly devised the rule “whenever you see a y in a term, stick on a y prime” (a more common mistake).

The main misconception is that the student is not interpreting y as a variable, and more importantly, a variable that is dependent on x. (And thus, because of that, the product and chain rules both come into play.)

Although I have only been teaching calculus for 5 years, I have found that it has gotten easier to teach kids the algebraic steps to do implicit differentiation, and they eventually get it. But it has been a much harder uphill battle to get them to understand (a) what exactly the meaning of what they are doing is and (b) what the connection is to everything else is. I’m getting better at it, but this throws so many abstract concepts at them that I can understand why it’s challenging.

Implicit differentiation, when I saw it first at school, left me feeling really uneasy. We had been shown differentiation of real functions and given some proofs about how that worked and what it meant. Then, without a warning, we see differentiation applied to something that is not a function. Huh?

Now, many years later, I know this was wrong. At the very least we should have been warned that something new was happening and told that a justification did exist, but was complicated, so for now all we had to do was learn the process.

My eldest son has recently been learning calculus and the teacher did exactly the same to him as was done to me, decades earlier.

It’s no wonder that students do not understand implicit differentiation well and just try to copy what they have seen, rather inaccurately.

The notation for calculus and the style of writing implicit differentiation could be improved, and so could the teaching of it. Why this has not happened in over 30 years I do not know.