Two students differentiate using different starting identities

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Two students differentiate tan⁡x\tan x using different starting identities. Student A writes tan⁡x=sin⁡xcos⁡x\tan x=\dfrac{\sin x}{\cos x} and differentiates. Student B writes tan⁡x=sec⁡x sin⁡x\tan x=\sec x\,\sin x and differentiates, where it is known that ddx(sec⁡x)=sec⁡xtan⁡x\dfrac{d}{dx}(\sec x)=\sec x\tan x. Both students obtain ddx(tan⁡x)=sec⁡2x\dfrac{d}{dx}(\tan x)=\sec^{2}x. (1) Describe, in words, why the two distinct algebraic representations of tan⁡x\tan x used by the students must lead to the same derivative even though the intermediate steps are different. (2) Explain how the equality of the two final results illustrates the consistency of the derivative with respect to equivalent algebraic forms of a function.

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