Summary Flashcards of Key Equations for Pure Mathematics A-Level (year 2)

TRIG DIFFERENTIATION DIFFERENTIATION DIFFERENTIATION DIFFERENTIATION
𝑎 sin 𝑥 ± 𝑏 cos 𝑥 →
𝑅 sin(𝑥 ± 𝑎) 𝑦 = ln 𝑥 𝑦 = 𝑎𝑘𝑥 𝑦 = 𝑎𝑥 Chain
𝑎 cos 𝑥 ± 𝑏 sin 𝑥 → 𝑑𝑦 1 𝑑𝑦 𝑑𝑦 𝑑𝑦 𝑑𝑦 𝑑𝑣
= ×1 = 𝑎𝑘𝑥 𝑘𝑙𝑛 𝑎 = 𝑎 𝑥 ln 𝑎 = ×
𝑅 cos (𝑥 ∓ 𝑎) 𝑑𝑥 𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑢 𝑑𝑥


DIFFERENTIATION DIFFERENTIATION DIFFERENTIATION DIFFERENTIATION DIFFERENTIATION
S
Product Quotient: 𝑓 (𝑥) = cos 𝑘𝑥 𝑓 (𝑥) = sin 𝑘𝑥
𝑑𝑦 𝑑𝑣 𝑑𝑢 𝑑𝑢 𝑑𝑣 -C C
=𝑢 + 𝑣 𝑑𝑦 𝑣 −𝑢 𝑓 ′(𝑥) = −𝑘𝑠𝑖𝑛 𝑘𝑥 𝑓 ′(𝑥) = 𝑘𝑐𝑜𝑠 𝑘𝑥
𝑑𝑥 𝑑𝑥 𝑑𝑥 = 𝑑𝑥 2 𝑑𝑥
𝑑𝑥 𝑣 -S

DIFFERENTIATION INTEGRATION INTEGRATION INTEGRATION VECTORS

Concave: 𝑓 ′′(𝑥) ≤ 0 Distance from origin to
Convex: 𝑓 ′′ (𝑥) ≥ 0 (𝑥, 𝑦, 𝑧) is
Point of inflection: 𝑓 ′′ (𝑥)
√𝑥 2 + 𝑦 2 + 𝑧 2
changes sign


VECTORS VECTORS

𝑝 𝑎 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘 makes
𝑝𝑖 + 𝑞𝑗 + 𝑟𝑘 = 𝑞 an angle with the positive
𝑥
𝑟 x-axis then 𝑐𝑜𝑠𝜃𝑥 = |𝑎|
Same for y and z