Page 28 - Ranger SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

Solution Example 4
(a) The gradient of the chord AB Find the first derivative of y = 2 using
(y + δy) – y x + 1
= (x + δx) – x the first principles.
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δy
= Solution
δx
2
(b) y = f(x) = 2x 2 y = f(x) = x + 1
δy = f(x + δx) – f(x) δy = f(x + δx) – f(x)
= 2(x + δx) – 2x 2 2 2
2
= 2[x + 2xδx + (δx) ] – 2x 2 = x + δx + 1 – x + 1
2
2
= 4xδx + 2(δx) 2 2(x + 1) – 2(x + δx + 1)
δy = 4xδx + 2(δx) 2 = (x + δx + 1)(x + 1)
δx δx –2δx
= 4x + 2δx = (x + δx + 1)(x + 1)
dy = lim δy δy –2
dx δx : 0 δx δx = (x + δx + 1)(x + 1)
lim
= δx : 0 (4x + 2δx) dy = lim δy
= 4x dx δx : 0 δx
–2
lim
= δx : 0 (x + δx + 1)(x + 1)
–2
Quiz = (x + 1)(x + 1)
Quiz
Quiz
Quiz
Find the value of lim 3x – 1 . = –2 2
x : 0 x (x + 1)

2.2 The First Derivative

1. Given that a, n are constants, u = f(x) and v = g(x).

Function y First derivative Function y First derivative
a 0 uv dv du
u dx + v dx
ax n anx n – 1 u du dv Form 5
v v dx – u dx
v 2
u + v du dv h(u) where dy du
dx + dx u = f(x) du × dx

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