Practice Flashcards
d/dx(tan x) = ?
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All Flashcards in Topic 5.15
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5.15.18 cards
d/dx(tan x) = ?
sec²x.
d/dx(sec x) = ?
sec x · tan x.
d/dx(csc x) = ?
−csc x · cot x (note the minus).
d/dx(cot x) = ?
−csc²x (note the minus).
d/dx(aˣ) = ?
aˣ ln a. (For a = e this is just eˣ, since ln e = 1.)
d/dx(log_a x) = ?
1/(x ln a). (For a = e this is 1/x.)
d/dx(arcsin x) and d/dx(arccos x)?
arcsin x → 1/√(1 − x²); arccos x → −1/√(1 − x²). Same fraction, opposite sign.
d/dx(arctan x) = ?
1/(1 + x²).
5.15.28 cards
How do you differentiate x² arctan x?
Product rule: u = x², v = arctan x ⇒ 2x·arctan x + x²·1/(1 + x²).
How do you differentiate eˣ sec x?
Product rule: eˣ sec x + eˣ sec x tan x = eˣ sec x (1 + tan x).
d/dx(arctan(3x)) = ?
Chain rule: 1/(1 + (3x)²) × 3 = 3/(1 + 9x²).
d/dx(tan(x²)) = ?
Chain rule: sec²(x²) × 2x = 2x sec²(x²).
d/dx(arcsin(2x)) = ?
Chain rule: 1/√(1 − (2x)²) × 2 = 2/√(1 − 4x²).
Which rule for y = (arctan x)/x?
Quotient rule: (u′v − uv′)/v² with u = arctan x, v = x.
Chain-rule trap for arcsin(2x) — what's the bottom?
√(1 − (2x)²) = √(1 − 4x²): square the WHOLE inner function, not just x.
How do you decide which rule to use?
Multiplied → product; divided → quotient; nested (one inside another) → chain.
Topic 5.15 study notes
Full notes & explanations for Further calculus (HL only)
Math AA exam skills
Paper structures, command terms & tips
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