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AP Calculus AB 2.8 The Product Rule - MCQs - Exam Style Questions

AP Calculus AB-MCQs and Answers - All Topics

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) \(\sec x\)
▶️ 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)

No-Calc Question

If \(f(x)=e^{2x}\,(x^{3}+1)\), then \(f'(2)=\;?\)

(A) \(6e^{4}\)
(B) \(21e^{4}\)
(C) \(24e^{4}\)
(D) \(30e^{4}\)

▶️ Answer/Explanation
Product rule:
\( f'(x)=\big(e^{2x}\big)’\,(x^{3}+1)+e^{2x}\,(x^{3}+1)’ \)
\( =2e^{2x}(x^{3}+1)+3x^{2}e^{2x} \)
\( =e^{2x}\!\left(2x^{3}+3x^{2}+2\right) \)
Evaluate at \(x=2\):
\(2(2^{3})+3(2^{2})+2=16+12+2=30\)
Hence \(f'(2)=30e^{4}\)
✅ Answer: (D) \(30e^{4}\)
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