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The prompt
The prompt
Mathematical inquiry processes: Reason; create examples; change representations. Conceptual field of inquiry: Rearrange equations; chain, product, and quotient rules; implicit differentiation.
Zelimir Jovicic devised the prompt for his year 13 A-level mathematics class during an Inquiry Maths workshop at Akademeia High School in Warsaw (Poland).
It is suitable for students who have learnt about the product and quotient rules but are yet to meet implicit differentiation.
The prompt is intriguing because it suggests that different derivatives can be derived from the same equation. The result of using the quotient rule to differentiate the first equation is different to the result of rearranging the equation and using the product rule.
Even though the use of 'however' in the prompt suggests there is a contradiction between the two parts, students might not notice that the equation has been rearranged. If that is the case, then the teacher should focus their attention on the two equations through questioning.
Question, notice and wonder
Question, notice and wonder
In the initial phase of the inquiry students have responded to the prompt in the following ways*:
The first equation has been rearranged to create the second one.
You cannot differentiate the equations because the first one is given in terms of the square of y and second one mixes the variables x and y on both sides.
Can you express y in terms of x for the first equation? If it is possible, then we could differentiate.
The derivatives are different depending on whether you use the quotient rule for the equation or the product rule for the rearranged equation. Is that possible?
If you use the quotient rule on an explicit equation and then rearrange the equation to use the product rule, do you always get different derivatives?
From the graph we can tell that the equation is not a function. When x = 1, y = 1 or -0.5.
The derivative at x = 1 is 0 when y = 1 (corresponding to the upper part of the curve) and -0.5 when y = -0.5 (lower part). Both derivatives give the same values at that point on the curve.
How do you differentiate an equation if the dependent variable is not isolated on one side?
Contradiction or not?
Contradiction or not?
Even though the two derivatives look different, it is possible to show they are equivalent by rearranging one to derive the other. This is only true for the relationship between x and y given in the equations - that is, for the values of x and y shown on the curve - and not, as some students have assumed, for any value of x and y.
October 2025
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* The questions and comments have been collated from two year 13 classes.
Creating the prompt
Creating the prompt
in the first draft (illustrated), Zelimir designed the prompt as a flow chart to draw out the differences between the derivatives.
Zelimir went on to plan the inquiry in four parts by (see the grid below):
Anticipating the questions his students would ask about the prompt;
Outlining the the type of reasoning the students would use;
Suggesting how the students would explore in more depth; and
Outlining the conclusions the students might reach.
By considering his students' abilities to reason mathematically and their level of experience with inquiry, Zelimir decided to use the questions and notes to orchestrate a guided inquiry with his year 13 class.
Lines of inquiry
Lines of inquiry
The lines of inquiry have been developed from students' comments about the prompt in the question, notice, and wonder phase.
1. Can you express y in terms of x for the first equation? If it is possible, then we could differentiate.
Students might suggest rearranging the equation using factorisation, square rooting both sides, or completing the square. The final suggestion leads to an expression for y in terms of x. Students can then differentiate using the chain rule (see the notes for the first line of inquiry).
2. If you use the quotient rule on an explicit equation and then rearrange the equation to use the product rule, do you always get different derivatives?
2. If you use the quotient rule on an explicit equation and then rearrange the equation to use the product rule, do you always get different derivatives?
Students generate their own examples involving algebraic fractions to show that the product and quotient rules (and, if the fraction can be written in a suitable form, the power rule) give the same derivative. Using the power rule on the equation in the table (right), y = 2x-1 + 3x-2 leads to a derivative of -2x-2 - 6x-3.
This line of inquiry might finish with a class discussion of the advantages and disadvantages of using each of the three methods.
3. How do you differentiate an equation if the dependent variable is not isolated on one side?
The teacher can use this question as an opportunity for flipped learning. Students research how to differentiate an implicit equation - that is, an equation in which the dependent variable is not isolated - in preparation for the next lesson.
Alternatively, the teacher might present the class with the model solutions (pictured) and ask students to 'think, pair, share' their understanding of implicit differentiation.
4. Generate examples using different types of equations.
Depending on the prior learning, students can generate examples using algebraic, logarithmic, trigonometric, and hyperbolic equations. To launch this line of inquiry, the class might co-construct an example under the teacher's guidance.
Students could also set their examples as questions to peers once they have shown the derivatives in their examples are equivalent.
(Solutions to the questions on the PowerPoint slide.)
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