Combining the new derivatives
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Flip to reveal answersHow do you differentiate x² arctan x?
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All 8 Flashcards — Combining the new derivatives
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Question
How do you differentiate x² arctan x?
Answer
Product rule: u = x², v = arctan x ⇒ 2x·arctan x + x²·1/(1 + x²).
Question
How do you differentiate eˣ sec x?
Answer
Product rule: eˣ sec x + eˣ sec x tan x = eˣ sec x (1 + tan x).
Question
d/dx(arctan(3x)) = ?
Answer
Chain rule: 1/(1 + (3x)²) × 3 = 3/(1 + 9x²).
Question
d/dx(tan(x²)) = ?
Answer
Chain rule: sec²(x²) × 2x = 2x sec²(x²).
Question
d/dx(arcsin(2x)) = ?
Answer
Chain rule: 1/√(1 − (2x)²) × 2 = 2/√(1 − 4x²).
Question
Which rule for y = (arctan x)/x?
Answer
Quotient rule: (u′v − uv′)/v² with u = arctan x, v = x.
Question
Chain-rule trap for arcsin(2x) — what's the bottom?
Answer
√(1 − (2x)²) = √(1 − 4x²): square the WHOLE inner function, not just x.
Question
How do you decide which rule to use?
Answer
Multiplied → product; divided → quotient; nested (one inside another) → chain.
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Topic 5.15 hub
Further calculus (HL only)
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