AP Calculus AB 2.8 The Product Rule - MCQs - Exam Style Questions
No-Calc Question
If \(f(x)=\sec x \tan x\), then \(f'(x)=\)
(A) \(\sec^{3}x+\sec x\,\tan^{2}x\)
(B) \(2\sec^{2}x\,\tan x\)
(C) \(\sec^{3}x\,\tan x\)
(D) \(\sec x\)
(B) \(2\sec^{2}x\,\tan x\)
(C) \(\sec^{3}x\,\tan x\)
(D) \(\sec x\)
▶️ Answer/Explanation
Product rule: \(\dfrac{d}{dx}[\sec x \tan x] = (\sec x)’ \tan x + \sec x \, (\tan x)’\).
Use \((\sec x)’=\sec x \tan x\) and \((\tan x)’=\sec^{2}x\).
Then \(f'(x)=\sec x \tan x \cdot \tan x + \sec x \cdot \sec^{2}x\).
Simplify: \(f'(x)=\sec x \tan^{2}x + \sec^{3}x\).
✅ Answer: (A)
Use \((\sec x)’=\sec x \tan x\) and \((\tan x)’=\sec^{2}x\).
Then \(f'(x)=\sec x \tan x \cdot \tan x + \sec x \cdot \sec^{2}x\).
Simplify: \(f'(x)=\sec x \tan^{2}x + \sec^{3}x\).
✅ Answer: (A)