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The prompt
The prompt
Mathematical inquiry processes: Generate examples; identify patters; generalise, and prove. Conceptual field of inquiry: Differentiation of trigonometric and hyperbolic functions; product rule; proof by mathematical induction.
The prompt originated in a question from a Core Pure examination paper for A-level Further Mathematics. The question required candidates to show that the fourth derivative of f(x) = sin(x) sinh(x) is -4f(x) (see below). The prompt generalises the result to all derivatives that are multiples of four.
Once students have identified the pattern for the first eight or twelve derivatives, they can reason that the generalisation is true.
Question, notice, and wonder
Question, notice, and wonder
The prompt is aimed at students who can differentiate trigonometric functions but have yet to meet hyperbolic functions. The familiar part of the function draws students into the inquiry, while the unfamiliar part leads to questions and a search for meaning.
One successful approach is flipped learning in which students study hyperbolic functions independently before they see the prompt. The question, notice, and wonder phase then becomes an opportunity for the teacher to assess the accuracy and depth of that knowledge.
Year 13 students who had undertaken home study before the inquiry responded to the prompt in the following ways:
'sinh' means hyperbolic sine, which is defined using exponentials.
You can differentiate the function using the product rule: f'(x) = cos(x) sinh(x) + sin(x) cosh(x).
The fourth derivative is -4 sin(x) sinh(x) according to the prompt.
If the prompt is true, the fourth derivative is simpler than the first derivative. It will only be true if the second and third derivatives contain negative terms.
If the prompt is true, how would we prove it?
Guided inquiry
Guided inquiry
Once the class is satisfied that the prompt is true, the teacher can orchestrate the co-construction of a proof by mathematical induction. Students then use the generalisation and proof as a model for their own examples. They create functions with two terms from different columns in the table below, identify a pattern in the higher-order derivatives (if there is one), and then form and prove a generalisation.
July 2026
Lines of inquiry
Lines of inquiry
The slides contain lines of inquiry based on six regulatory cards. In a guided inquiry, students suggest how the inquiry could proceed by selecting a card and justifying their selection to the class. For a detailed description of the meaning of each card in the context of the inquiry, see the lines of inquiry below.
Students who select this card are likely to require an explanation about hyperbolic functions and their derivatives. The teacher should define the hyperbolic sine, relate the definition to a graphical representation, and draw out from the class how to find its derivative (assuming the class is familiar with differentiating exponential functions). Individuals might then take the same approach with the hyperbolic cosine and tangent.
Students who select this card will feel confident about using the derivatives of hyperbolic functions and want to work out successive derivatives. The teacher might pose the following questions:
How many higher-order derivatives do you require to decide if the prompt is true or false?
Do you need to work out all the derivatives from the one before?
Is there a pattern you can use to justify your conclusion about the prompt?
Students use their knowledge of mathematical induction to collaborate in constructing a proof.
Students follow the same process with the product of their own functions. The teacher guides students to create examples by selecting from exponential, trigonometric, and hyperbolic functions. Students initially use two functions (see example below) before exploring combinations of three.
The aim with each example is to form a generalisation. That is possible when the derivatives develop in a predictable pattern, such as with e2xcosh(x).
However, as students discover, higher order derivatives do not always give rise to a consistent pattern. For example, the first four derivatives of e2xcos(x) are:
f'(x) = e2x(2cos(x) - 1sin(x))
f''(x) = e2x(3cos(x) - 4sin(x))
f'''(x) = e2x(2cos(x) - 11sin(x))
f'v(x) = e2x(-7cos(x) - 24sin(x))
Students use mathematical induction again to prove their own generalisations.
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