Class notes Basic Calculus

BASIC CALCULUS REVIEWER
Differential Calculus



LESSON: DIFFERENTIATION RULES RULE 3: Constant Multiple Rule of
Differentiation
● the process of finding the
DERIVATIVE of a function
● 𝑘 · 𝑔(𝑥) = 𝑘 · 𝑔'(𝑥)
[k is a constant]

RULE 1: Constant Rule of Differentiation ● Ex. 1
𝑓(𝑥) = 3x2 [k = 3 , g(x) = x2]
● constant = 0
𝑓'(𝑥) = k ∘ g’(x)
● Ex. 1 𝑓'(𝑥) = 3 ∘ Dx (x2) → [refer on rule 2]
𝑓(𝑥) = 61, 153 𝑓'(𝑥) = 3 ∘ 2x
𝑓'(𝑥) = 6x
☑️
𝑓'(𝑥) = 0
☑️
● Ex. 2 ● Ex. 2
𝑓(𝑥) = 1, 000, 000 𝑓(𝑥) = 12x4 [k = 12 , g(x) = x4]
𝑓'(𝑥) = k ∘ g’(x)
☑️
𝑓'(𝑥) = 0
𝑓'(𝑥) = 12 ∘ Dx (x4) →[refer on rule 2]
𝑓'(𝑥) = 12 ∘ 4x3
𝑓'(𝑥) = 48x3
RULE 2: Power Rule of Differentiation ☑️
● xn = nxn-1 [n is a positive integer]
RULE 4: Sum Rule of Differentiation
● Ex. 1
𝑓(𝑥) = x4 [n = 4] ● ℎ(𝑥) ± 𝑔(𝑥) = ℎ'(𝑥) ± 𝑔'(𝑥)
𝑓'(𝑥) = 4x4-1 [if h’(x) and g’(x) exist]
𝑓'(𝑥) = 4x3
☑️ ● Ex. 1
𝑓(𝑥) = 3x² - 7x [h(x) = 3x² , g(x) = 7x]
● Ex. 2 𝑓'(𝑥) = Dx (3x²) - Dx (7x) →[refer on
𝑓(𝑥) = x15 [n = 15] rule 3]
𝑓'(𝑥) = 6x - 7
𝑓'(𝑥) = 15x15-1
𝑓'(𝑥) = 15x14 ☑️
☑️
● Ex. 2