The algebraic techniques module4

1. The definition of derivative can lead us to the derivation of the rules of differentiation or the algebraic
techniques of differentiation.
ALGEBRAIC TEHCNIQUES OF DIFFERENTIATION
ALGEBRAIC TEHCNIQUES OF
DIFFERENTIATION
EXAMPLES
A. Derivative of a Constant
If 𝑓(𝑥) = 𝑘, where 𝑘 is a real
number, then 𝑓′(𝑥) = 0
𝑓(𝑥) = 3
𝒇′(𝒙) = 𝟎
B. Derivative of a Power Function
(Power Rule)
If 𝑓(𝑥) = 𝑥𝑛
, where 𝑛 is a real
number, then 𝑓′(𝑥) = 𝑛𝑥𝑛−1
𝑓(𝑥) = 𝑥2
𝑓(𝑥) = 2𝑥2−1
𝒇(𝒙) = 𝟐𝒙
C. Derivative of the Sum or
Difference of Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) = 𝑓(𝑥) ±
𝑔(𝑥) then 𝑝′(𝑥) = 𝑓′(𝑥) ± 𝑔′(𝑥)
𝑓(𝑥) = 3𝑥4
− 2𝑥3
+ 4𝑥2
− 12𝑥 − 5
𝑓′(𝑥) = (4)(3)(𝑥4−1) − (3)(2)(𝑥3−1) + (2)(4)(𝑥1−1) − 0
𝒇′(𝒙) = 𝟏𝟐𝒙𝟑
− 𝟔𝒙𝟐
+ 𝟖𝒙
D. Derivative of the Product of Two
Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) = 𝑓(𝑥) ∙
𝑔(𝑥) then 𝑝′(𝑥) = [𝑓(𝑥) ∙ 𝑔′(𝑥)] ∙
[𝑓′(𝑥)𝑔(𝑥)]
𝑓(𝑥) = (2𝑥2
+ 1)(4𝑥2
+ 2)
𝑓′(𝑥) = (2𝑥2
+ 1)[(4)(2)(𝑥2−1)] + [(2)(2)(𝑥2−1)](4𝑥2
+ 2)
𝑓′(𝑥) = (2𝑥2
+ 1)(8𝑥) + (4𝑥)(4𝑥2
+ 2)
𝑓′(𝑥) = 16𝑥2
+ 8𝑥 + 16𝑥2
+ 8𝑥
𝑓′(𝑥) = 32𝑥2
+ 16𝑥
E. Derivative of the Quotient of
Two Functions
If 𝑓 and 𝑔 are functions and 𝑝 is a
function defined by 𝑝(𝑥) =
𝑓(𝑥)
𝑔(𝑥)
where 𝑔(𝑥) ≠ 0, then
𝑝′(𝑥) =
𝑔(𝑥)𝑓′(𝑥)−𝑓(𝑥)𝑔′(𝑥)
[𝑔(𝑥)]2
𝑓(𝑥) =
2𝑥 − 3
4𝑥 + 1
𝑓′(𝑥) =
(4𝑥 + 1)(2) − (2𝑥 − 3)(4)
(4𝑥 + 1)2
𝑓′(𝑥) =
(8𝑥 + 2) − (8𝑥 − 12)
(4𝑥 + 1)(4𝑥 = 1)
𝑓′(𝑥) =
14
16𝑥2 + 8𝑥 + 1