Page 26 - Ranger SPM 2022 - Additional Mathematics

CHAPTER 2 DIFFERENTIATION Form 5
Additional Mathematics SPM Chapter 2 Differentiation

Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
Learning Area: Calculus

CONCEPT MAP

Differentiation

First principles
dy lim δy Application of differentiation
dx = δx : 0 δx (a) If it is given a curve y = f(x),
lim
= δx : 0 f(x + δx) – f(x) then
(i) Equation of tangent at a point
δx
(x 1 , y 1 ) dy
First derivative y - y 1 = dx (x - x 1 )
dy (ii) Equation of normal at a point
(i) If y = ax , then dx = anx n – 1 (x 1 , y 1 )
n
d 1
(ii) dx [f(x) ± g(x)] y - y 1 = – dy (x - x 1 )
d
d
dx
= dx [f(x)] ± dx [g(x)] (b) Turning point is a maximum point
2
(iii) Chain rule dy = 0 and d y 2  0
dx
dx
dy = dy × du (c) Turning point is a minimum point
dx
du
dx
2
(iv) Multiplication rule dy d y
d dv du dx = 0 and dx 2  0
dx (uv) = u dx + v dx (d) Point of inflection
2
(v) Quotient rule dv dy = 0 and d y 2 = 0 Form 5
du
dx
dx
d u v dx – u dx (e) Small change and approximation
( ) =
dx v v 2 dy
δy  dx × δx
f(x + δx)  y + dy δx or
Second derivative dx dy
d y d dy d f(x + δx)  f(x) + dx δx
2
dx 2 = ( ) or f″(x) = dx [f′(x)]
dx dx
179
179
02 Ranger Mate Tambahan Tg5.indd 179 25/02/2022 9:23 AM