When is differentiated

MATHEMATICS · Single correct · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

When sec⁡x\sec x is differentiated, the result is sec⁡xtan⁡x\sec x\tan x. Which of the following best accounts for the appearance of the factor tan⁡x\tan x? (a) Differentiating sec⁡x\sec x requires the product rule on sin⁡x\sin x and cos⁡x\cos x, producing an extra tan⁡x\tan x factor. (b) Writing sec⁡x=(cos⁡x)−1\sec x=(\cos x)^{-1} and applying the chain rule multiplies −sin⁡x-\sin x by cos⁡−2x\cos^{-2}x; the ratio −sin⁡x/cos⁡x-\sin x/\cos x simplifies to tan⁡x\tan x, giving sec⁡xtan⁡x\sec x\tan x. (c) Applying the chain rule to (cos⁡x)−1(\cos x)^{-1} gives −(cos⁡x)−2 (−tan⁡x)-(\cos x)^{-2}\,(-\tan x) because the derivative of cos⁡x\cos x is −tan⁡x-\tan x; simplifying then supplies the needed tan⁡x\tan x factor. (d) Because tan⁡x\tan x is the reciprocal of sec⁡x\sec x, it is inserted to keep the derivative dimensionless.

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