Page 35 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation
(c) Let y = f(x), 2. Find the limit for each of the following when x → 0.
that is y = 3x – 8x (a) (b) y
2
y + dy = 3(x + dx) – 8(x + dx) y
2
3x – 8x + dy = 3[x + 2xdx + (dx) ] – 8x – 8dx O x
2
2
2
3x – 8x + dy = 3x + 6xdx + 3(dx) – 8x – 8dx O x
2
2
2
dy = 6xdx + 3(dx) – 8dx y = –3x – 2 y = 3 – x 3
2
2
dy = 6x + 3(dx) – 8
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dx (c) 3 x + 5 (d) 2 – x
2
2 3 + x
lim dy
2
\ dx → 0 dx = 6x + 0 – 8 (e) 4x – 3x
5x
and dy = 6x – 8 3. Find the values of each of the following.
dx
(a) lim (3 – 2x + x )
2
x → 0
REMEMBER! (b) x → 0 (2 – x)(5 – x)
lim
4
Each of the x term must be added with dx because each (c) lim x + 5
of them experience a small change. x → 0 x – 10
2
(d) lim x + 8
x → 0 (x – 2)(x + 4)
lim x – 5x
(d) Let y = f(x), (e) x → 0 3 4x 3 4
that is y = 6 lim x – 16
2
x (f) x → 0 x – 4
y + dy = 6
x + dx 4. Differentiate each of the following function by using
6 + dy = 6 first principle.
x x + dx (a) f(x) = 6x (b) f(x) = 1 – 2x 2
1
6
2
dy = 6 – (c) f(x) = (2x – 1) (d) f(x) = – x
x + dx x 5. Determine the first derivative of each of the following
dy = 6x – 6(x + dx) functions by using first principle.
(x + dx)(x) (a) y = 4(1 – 3x) (b) y = 6x – 2
2
2
dy = 6x – 6x – 6dx x
(x + dx)(x) (c) y = – 2 (d) y = 5
dy = –6 3 8x
dx (x + dx)(x)
lim dy
–6
\ dx → 0 dx = (x + 0)(x) 2.2 The First Derivative
and dy = 6
dx x 2
A Deriving the formula of first
Form 5
derivative inductively for the
Try Questions 4 – 5 in ‘Try This! 2.1’
function y = ax n
Try This! 2.1 1. The first derivative differentiation involves the
dy
process of obtaining from y or f'(x) from f(x).
dx
1. Determine the limit value of each of the following 2. If y = ax where a and n are constants, then
n
by using (i) table, (ii) graph, (iii) direct substitution
method.
lim
n
(a) x → 0 (8x) dy = dy [ax ] = f'(x) = nax n – 1
dx
dx
(b) lim (x + 3)
2
x → 0 Observe that this formula is obtained inductively.
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