WISKUNDE
GRAAD 12
NOG OEFENINGE
Afgeleides - gebruik die reëls : antwoorde.
MATHEMATICS
GRADE 12
MORE EXERCISES
Derivatives - use the rules : answers.
Vraag / Question 1
dy
1.1 ─── = (3)x
3 − 1
dx
= 3x
2
V/Q. 1.1
dy
1.3 ─── = (7)x
7 − 1 + (3)x
3 − 1
dx
= 7x
6 + 3x
2
V/Q. 1.3
dy
1.5 ─── = (4)x
4 − 1 − (2)x
2 − 1
dx
= 4x
3 − 2x
V/Q. 1.5
dy
1.7 ─── = 2(3)x
3 − 1 + 3(2)x
2 − 1
dx
= 6x
2 + 6x
V/Q. 1.7
dy
1.9 ─── = 5(4)x
4 − 1 − 4(3)x
3 − 1 + 8(1)x
1 − 1
dx
= 20x
3 − 12x
2 + 8
V/Q. 1.9
dy
1.11 ─── = 7(3)x
2 − 3(2)x + 2(1)
dx
= 21x
2 − 6x + 2
V/Q. 1.11
__
1.13 y = √x = x
1/2
dy
─── = (½)x
½ − 1 = ½x
−½
dx
1
= ───
V/Q. 1.13
2√x
___
1.15 y = √2x
3 = √2 × x
3/2
dy √2 3 3√2
─── = ── × ─ × x
3/2 − 1 = ─── × x
½
dx 1 2 2
___
3√2x
= ──────
V/Q. 1.15
2
4
1.17 y = ── = 4x
−1
x
dy
─── = 4(−1)x
−2 = −4 x
−2
dx
4
= ─ ───
V/Q. 1.17
x
2
6
1.19 y = ── = 6x
−3
x
3
dy
─── = 6(−3)x
−4 = −18 x
−4
dx
18
= ─ ───
V/Q. 1.19
x
4
x
6 x
4
1.21 y = ── ─ ──
9 8
dy
1 6 1 4
─── = ── × ── × x
5 ─ ── × ── × x
3
dx
9 1 8 1
2x
5 x
3
= ─── ─ ───
V/Q. 1.21
3 2
x
5 2x
3
1.23 y = ── ─ ──
15 6
dy
1 5 2 3
─── = ── × ── × x
4 ─ ── × ── × x
2
dx
15 1 6 1
x
4
= ─── ─ x
2
V/Q. 1.23
3
3
1.25 y = ─── = 3x
−1/2
√x
dy
─── = 3(−½)x
−3/2 = −(3/2) x
−3/2
dx
3
= ─ ───
V/Q. 1.25
2√x
3
−3
1.27 y = ─── − x
10 = ─ 3x
−½ − x
10
√x
dy
─── = ─3(−½)x
−3/2 ─ 10x
9
dx
3
= ─── − 10x
9
V/Q. 1.27
2√x
3
___
√3x
4 5
1.29 y = ─── − ───
= ⅓(3x
4)
½ − (5/3)x
−5
3 3x
5
dy
─── =
⅓(3
½)(2)x
1 ─ (5/3)(−5)x
−6
dx
2√3 x 25
= ───── + ───
V/Q. 1.29
3 3x
6
√x − 4
1.31 y = ──────
= 1 − 4x
−½
√x
dy
─── =
0 ─ 4(−½)x
−3/2
dx
2
= ────
V/Q. 1.31
√x
3
3x + 8
1.33 y = ──────
= 3x
½ + 8x
−½
√x
dy
─── =
3(½)x
−½ + 8(−½)x
−3/2
dx
3 4
= ──── − ────
V/Q. 1.33
2√x √x
3
x
2 − 3x − 2
1.35 y = ──────
= ½x − (3/2) − x
−1
2x
dy
─── =
½(1)x
0 − 0 − (−1)x
−2
dx
1 1
= ─── + ────
2 x
2
V/Q. 1.35
1.37 y = (x − 1)(x + 5) = x
2 + 4x − 5
dy
─── = (2)x
1 + 4(1)x
0 − 0
dx
= 2x + 4
V/Q. 1.37
1.39 y = (2x + 1)(x + 3) = 2x
2 + 7x + 3
dy
─── = 2(2)x
1 + 7(1)x
0 − 0
dx
= 4x + 7
V/Q. 1.39
1.41 y = (2x + 3)(4x + 5) = 8x
2 + 22x + 15
dy
─── = 8(2)x
1 + 22(1)x
0 − 0
dx
= 16x + 22
V/Q. 1.41
1.43 y = (x + 2)
2 = x
2 + 4x + 4
dy
─── = (2)x
1 + 4(1)x
0 − 0
dx
= 2x + 4
V/Q. 1.43
1.45 y = (x + 4)
2 = x
2 + 8x + 16
dy
─── = (2)x
1 + 8(1)x
0 − 0
dx
= 2x + 8
V/Q. 1.45
1.47 y = (2x
3 − 3)(3x
4 + 3) = 6x
7 − 9x
4 + 6x
3 − 9
dy
─── = 6(7)x
6 − 9(4)x
3 + 6(3)x
2 − 0
dx
= 42x
6 − 36x
2 + 18x
2
V/Q. 1.47
1.49 y = (x − 3)(x + 2)(x + 5) = x
3 + 4x
2 − 11x − 30
dy
─── = (3)x
2 − 4(2)x − 11(1)x
0 − 0
dx
= 3x
2 − 8x − 11
V/Q. 1.49
6x
5 + 4x
3 − 8x
2 + 10x
1.51 y = ────────────────
2x
= 3x
4 + 2x
2 − 4x + 5 x ≠ 0
dy
─── = 3(4)x
3 + 2(2)x
1 − 4
dx
= 12x
3 + 4x − 4
V/Q. 1.51
2x
4 − 8x
3 + 4x
2
1.53 y = ───────────
2x
2
= x
2 − 4x + 2 x ≠ 0
dy
─── = (2)x
1 − (4)x
0 − 0
dx
= 2x − 4
V/Q. 1.53
x
2 − 9
1.55 y = ───────
x + 3
(x − 3)(x + 3)
= ─────────
x + 3
= x − 3 (x + 3) ≠ 0
dy
─── = x
0 − 0
dx
= 1
V/Q. 1.55
x
2 − 25
1.57 y = ──────
x + 5
(x − 5)(x + 5)
= ─────────
x + 5
= x − 5 (x + 5) ≠ 0
dy
─── = x
0 − 0
dx
= 1
V/Q. 1.57
9x
2 − 25
1.59 y = ────────
3x − 5
(3x − 5)(3x + 5)
= ─────────
3x − 5
= 3x + 5 (3x − 5) ≠ 0
dy
─── = 3(1)x
0 − 0
dx
= 3
V/Q. 1.59
x
2 − 8x + 12
1.61 y = ───────────
x − 2
(x − 2)(x − 6)
= ─────────
x − 2
= x − 6 (x − 2) ≠ 0
dy
─── = 1x
0 − 0
dx
= 1
V/Q. 1.61
x
2 − 8x + 15
1.63 y = ───────────
x − 5
(x − 3)(x − 5)
= ─────────
x − 5
= x − 3 (x − 5) ≠ 0
dy
─── = 1x
0 − 0
dx
= 1
V/Q. 1.63
4x
2 + 8x + 3
1.65 y = ───────────
2x + 5
(2x + 1)(2x + 3)
= ─────────
2x + 1
= 2x + 3 (2x + 1) ≠ 0
dy
─── = 2(1)x
0 + 0
dx
= 2
V/Q. 1.65
dy
1.2 ─── = (8)x
8 − 1
dx
= 8x
7
V/Q. 1.2
dy
1.4 ─── = (5)x
5 − 1 + (2)x
2 − 1
dx
= 5x
4 + 2x
V/Q. 1.4
dy
1.6 ─── = (3)x
3 − 1 − 3(2)x
2 − 1
dx
= 3x
2 − 6x
V/Q. 1.6
dy
1.8 ─── = 4(5)x
5 − 1 − 2(3)x
3 − 1
dx
= 20x
4 − 6x
2
V/Q. 1.8
dy
1.10 ─── = 4(5)x
5 − 1 − 2(3)x
3 − 1
dx
= 20x
4 − 6x
2
V/Q. 1.10
dy
1.12 ─── = 4(3)x
2 − 3(2)x + 2(1)
dx
= 12x
2 − 6x + 2
V/Q. 1.12
___
1.14 y = √4x
3 = √4 × x
3/2
dy 2 3
─── = ─ × ─ × x
3/2 − 1 = 3x
½
dx 1 2
= 3√x
V/Q. 1.14
___
1.16 y = √3x
5 = √3 × x
5/2
dy √3 5
─── = ── × ─ × x
5/2 − 1 = 5(√3)x
3/2
dx 1 2
___
= 5√3x
3
V/Q. 1.16
8
1.18 y = ── = 8x
−2
x
2
dy
─── = 8(−2)x
−3 = −16 x
−3
dx
16
= ─ ───
V/Q. 1.18
x
3
7
1.20 y = ── = 7x
−5
x
5
dy
─── = 7(−5)x
−6 = −35 x
−6
dx
35
= ─ ───
V/Q. 1.20
x
6
x
3 x
2
1.22 y = ── ─ ──
4 6
dy
1 3 1 2
─── = ── × ── × x
3 ─ ── × ── × x
dx
4 1 6 1
3x
3 x
= ─── ─ ───
V/Q. 1.22
4 3
3x
4 5x
3
1.24 y = ── ─ ──
2 3
dy
3 4 5 3
─── = ── × ── × x
3 ─ ── × ── × x
2
dx
2 1 3 1
= 6x
3 ─ 5x
2
V/Q. 1.24
1
1.26 y = ─── = ½x
−1/2
2√x
dy
─── = ½(−½)x
−3/2 = −¼ x
−3/2
dx
1
= ─ ───
V/Q. 1.26
4√x
3
√x 1
1.28 y = ─── − ─── = ⅓x
½ − (1/7)x
−4
3 7x
4
dy
─── = ⅓(½)x
−½ ─ (1/7)(−4)x
−5
dx
1 4
= ─ ─── + ───
V/Q. 1.28
6√x 7x
5
___
√x
5 9
1.30 y = ─── − ─── = 2x
5/2 − (9/8)x
−6
0,5 8x
6
dy
─── = 2(5/2)x
3/2 ─ (9/8)(−6)x
−7
dx
27
= 5√x
3 + ───
V/Q. 1.30
4x
7
4 x
2
1.32 y = ─── ─ ─── = 4x
−½ − (1/9)x
2
√x 9
dy
─── = 4(−½)x
−3/2 − (1/9)(2)x
1
dx
2 2x
= ─── − ───
V/Q. 1.32
√x
3 9
x
3 6
1.34 y = ─── ─ ─── = (1/6)x
3 − x
−2
6 x
2
dy
─── = (1/6)(3)x
2 − (−2)x
−3
dx
2
= ½x
2 + ───
V/Q. 1.34
x
3
1 − 2x + √x
1.36 y = ──────────
= x
−2 − 2x
−1 + x
−3/2
x
2
dy
─── =
(−2)x
−3 − 2(−1)x
−2 + (−3/2)x
−5/2
dx
2 2 3
= ─ ─── + ──── − ────
x
3 x
2 2√x
5
V/Q. 1.36
1.38 y = (x − 3)(x + 2) = x
2 − x − 6
dy
─── = (2)x
1 − (1)x
0 − 0
dx
= 2x − 1
V/Q. 1.38
1.40 y = (3x − 4)(x + 5) = 3x
2 + 11x − 20
dy
─── = 3(2)x
1 + 11(1)x
0 − 0
dx
= 6x + 11
V/Q. 1.40
1.42 y = (2x − 3)(5x + 2) = 10x
2 − 11x − 6
dy
─── = 10(2)x
1 − 11(1)x
0 − 0
dx
= 20x − 11
V/Q. 1.42
1.44 y = (x − 3)
2 = x
2 − 6x + 9
dy
─── = (2)x
1 − 6(1)x
0 + 0
dx
= 2x − 6
V/Q. 1.44
1.46 y = (3x + 2)
2 = 9x
2 + 12x + 4
dy
─── = 9(2)x
1 + 12(1)x
0 + 0
dx
= 18x + 12
V/Q. 1.46
1.48 y = (4x
2 − 5)(3x
4 + 4) = 12x
6 − 15x
4 + 16x
2 − 20
dy
─── = 12(6)x
5 − 15(4)x
3 + 16(2)x
1 − 0
dx
= 72x
5 − 60x
3 + 32x
V/Q. 1.48
1.50 y = (3x
3 − 7)(5x
4 + 3) = 15x
7 − 35x
4 + 9x
3 − 21
dy
─── = 15(7)x
6 − 35(4)x
3 + 9(3)x
2 − 0
dx
= 105x
6 − 140x
3 + 27x
2
V/Q. 1.50
9x
7 − 18x
5 + 6x
3 − 24x
1.52 y = ────────────────
3x
= 3x
6 − 6x
4 + 2x
2 − 8 x ≠ 0
dy
─── = 3(6)x
5 − 6(4)x
3 + 2(2)x
dx
= 18x
5 − 24x
3 + 4x;
V/Q. 1.52
4x
3 + 8x
2 − 12x
1.54 y = ────────────
4x
= x
2 + 2x − 3 x ≠ 0
dy
─── = (2)x
1 + (2)x
0 + 0
dx
= 2x + 2
V/Q. 1.54
x
2 − 16
1.56 y = ───────
x − 4
(x − 4)(x + 4)
= ─────────
x − 4
= x + 4 (x − 4) ≠ 0
dy
─── = x
0 + 0
dx
= 1
V/Q. 1.56
4x
2 − 9
1.58 y = ───────
2x − 3
(2x − 3)(2x + 3)
= ─────────
2x − 3
= 2x + 3 (2x − 3) ≠ 0
dy
─── = 2x
0 + 0
dx
= 2
V/Q. 1.58
25x
2 − 16
1.60 y = ─────────
5x + 4
(5x − 4)(5x + 4)
= ─────────
5x + 4
= 5x − 4 (5x + 4) ≠ 0
dy
─── = 5x
0 + 0
dx
= 5
V/Q. 1.60
x
2 + 5x + 6
1.62 y = ──────────
x + 2
(x + 2)(x + 3)
= ─────────
x + 2
= x + 3 (x + 2) ≠ 0
dy
─── = x
0 + 0
dx
= 1
V/Q. 1.62
x
2 − 6x − 7
1.64 y = ──────────
x + 1
(x − 7)(x + 1)
= ─────────
x + 1
= x − 7 (x + 1) ≠ 0
dy
─── = x
0 + 0
dx
= 1
V/Q. 1.64
6x
2 − 7x − 5
1.66 y = ───────────
5 − 3x
(3x − 5)(2x + 1)
= ───────────
−(3x − 5)
= −2x − 1 (3x − 5) ≠ 0
dy
─── = −2x
0 − 0
dx
= −2
V/Q. 1.66
Antwoorde / Answers 2
2.1 f(x) = x
4
f'(x) = 4x
3
V/Q. 2.1
2.3 f(x) = 3x
4 + 1
f'(x) = 4(3)x
3 = 12x
3
V/Q. 2.3
2.5 f(x) = 3x
6 + 2x
3 − 9x − 4
f'(x) = 18x
5 + 6x
2 − 9
V/Q. 2.5
2.7 f(x) = √(3x)
= √3 × x
½
f'(x) = (√3)(½)x
−½
√3
= ───
V/Q. 2.7
2√x
8
2.9 f(x) = ───
= 8 x
−2
x
2
f'(x) = (8)(−2)x
−3
16
= ─ ───
V/Q. 2.9
x
3
6
2.11 f(x) = ───
= 6x
−3
x
3
f'(x) = (6)(−3)x
−4
18
= ─ ───
V/Q. 2.11
x
4
1
2.13 f(x) = ───
= ½x
−½
2√x
f'(x) = (½)(−½)x
−3/2
1
= ─ ───
V/Q. 2.13
4√x
3
4√x − x
2.15 f(x) = ───────
= 4 − x
½
√x
f'(x) = 0 − (½)x
−½
1
= ─ ───
V/Q. 2.15
√x
1
2.17 f(x) = ─── + √x
= x
−½ + x
½
√x
f'(x) = (−½)x
−3/2 + (½)x
−½
1 1
= ─ ─── + ───
V/Q. 2.17
2√x
3 2√x
2.19 f(x) = (x − 6)
2
= x
2 − 12x + 36
f'(x) = 2x − 12
V/Q. 2.19
2.21 f(x) = (2x − 3)
2
= 4x
2 − 12x + 9
f'(x) = 8x − 12
V/Q. 2.21
2.23 f(x) = (4x + 3)
3
= 16x
2 + 24x + 9
f'(x) = 32x + 24
V/Q. 2.23
x
3 − 4x
2 − 3x + 18
2.25 f(x) = ─────────────
x + 2
(x + 2)(x
2 − 6x + 9)
= ─────────────
x + 2
= x
2 − 6x + 9
f'(x) = 2x − 6
V/Q. 2.25
2.2 f(x) = 2x
3
f'(x) = 6x
2
V/Q. 2.2
2.4 f(x) = 2x
5 − 6
f'(x) = 2(5)x
4 = 10x
4
V/Q. 2.4
2.6 f(x) = x
5 + 4x
3 − 7x
2 − 6x + 12
f'(x) = 5x
4 + 12x
2 − 14x − 6
V/Q. 2.6
2.8 f(x) = √x
3
= x
3/2
f'(x) = (3/2)x
½
3√x
= ───
V/Q. 2.8
2
3
2.10 f(x) = ─── = (3/2)x
−3
2x
3
f'(x) = (3/2)(−3)x
−2
9
= ─ ────
V/Q. 2.10
2x
2
3
2.12 f(x) = ───
= 3x
−½
√x
f'(x) = 3(−½)x
−3/2
3
= ─ ────
V/Q. 2.12
2√x
3
−3
2.14 f(x) = ─── − x
10
= −3x
−½ − x
10
√x
f'(x) = −3(−½)x
−3/2 − 10x
9
3
= ──── − 10x
9
V/Q. 2.14
2√x
3
2 − x
2.16 f(x) = ───── − ∛x = (2/5)x
−½ − (1/5)x
½ − x
⅓
5√x
f'(x) = (2/5)(−½)x
−3/2 − (1/5)(½)x
−½ − ⅓x(−⅔)
= ─ (⅕)x
−3/2 − (1/10)x
−½ − (⅓)x
−⅔
1 1
1
= ─ ─── ─ ─── ─ ────
V/Q. 2.16
5√x
3 10√x
3 ∛(x
2)
4
2.18 f(x) = (3√x + ───)
2
√x
2 × 3 × 4 × √x 4
2
= (3√x)
2 + ────────── + ────
√x
(√x)
2
16
= 9x + 24 + ─── = 9x + 24 + 16x
−1
x
f'(x) = 9(1) + 0 + 16(−1)x
−2
16
= 9 − ───
V/Q. 2.18
x
2
2.20 f(x) = (4x − 5)
2
= 16x
2 − 40x + 25
f'(x) = 32x − 40
V/Q. 2.20
2.22 f(x) = (3x − 4)
2
= 9x
2 − 24x + 16
f'(x) = 18x − 24
V/Q. 2.22
2.24 f(x) = (2x − 3)
3
= (2x − 3)(4x
2 − 12x + 9)
= 8x
3 − 36x
2 + 54x − 27
f'(x) = 24x
2 − 72x + 54
V/Q. 2.24
8x
3 − 1 (2x − 1)(4x
2 + 2x + 1)
2.26 f(x) = ─────── = ──────────────
2x − 1
2x − 1
= 4x
2 + 2x + 1
f'(x) = 8x + 2
V/Q. 2.26