Page 31 - Ranger SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

2.3 The Second Derivative

dy d d dy d y
2
Penerbitan Pelangi Sdn Bhd. All Rights Reserved.
1. Given y = f(x), then dx = dx [f(x)] is 2. dx dx dx 2
  can be written as
the first derivative. or fʺ(x) which also known as the
dy
Differentiating dx with respect to x second derivative.
d dy d d y d dy d
2
will give   or dx [fʹ(x)]. dx 2 =   or fʺ(x) = dx [fʹ(x)]
dx dx
dx dx
2
Example 10 d y 2 = –14(–3)(x – 2) –3 –1 (1)
d y dx = 42(x – 2) –4
2
Find for each of the following and
dx 2 2 = 42
hence, determine whether d y 2 is the same as (x – 2) 4
dx
dy
dy
–14
196
  2 .    (x – 2) 3 2 = (x – 2) 6
2
=
dx
dx
2x 3 1 d y dy 2
2
(a) y = – Then, 2  
≠
3 x dx dx
7
(b) y =
(x – 2) 2
Example 11
Solution d y dy
2
Given y = 3x(x – 1), express dx 2 + dx
2
2
(a) y = x – x –1 in terms of x. Hence, find the possible
3
3 d y dy
2
dy 2 values of x if + = 24.
= (3)x 3 – 1 – (–1)x –1 – 1 dx 2 dx
dx 3
= 2x + x –2 Solution
2
d y y = 3x – 3x
2
3
= 2(2)x 2 – 1 + (–2)x –2 – 1
dx 2 dy
2
= 4x – 2x –3 dx = 9x – 3
= 4x – 2 d y
2
x 3 dx 2 = 18x
dy
2
   2 x 1 2 2 d y 2 + dy = 18x + 9x – 3
= 2x +
2
dx
2
Form 5 Then, d y 2   2 dx d y + = 9x + 18x – 3
dx
2
dy
2
≠
dy
2
dx
dx
= 24
dx
dx
2
(b) y = 7(x – 2) –2 9x + 18x – 3 = 24
2
2
dy = 7(–2)(x – 2) –2 – 1 (1) 9x + 18x – 27 = 0
2
dx x + 2x – 3 = 0
= –14(x – 2) –3 (x + 3)(x – 1) = 0
= –14 x = –3 or 1
(x – 2) 3
184
02 Ranger Mate Tambahan Tg5.indd 184 25/02/2022 9:23 AM