Jika kita memiliki fungsi f(x) = g(h(x)) maka belaku

atau

sehingga
f ‘(x) = g’ (h(x)). h'(x)
Kasimpulan
f(x) = g(h(x)) → f ‘(x) = g’ (h(x)). h'(x)
f(x) = g(h(k(x))) → f ‘(x) = g’ (h(k(x))). h'(k(x)).k'(x)
f(x) = g(h(k(m(x))))
→ f ‘(x) = g'(h(k(m(x)))).h'(k(m(x))).k'(m(x)).m'(x)
Konsekuensi
f(x) = hn(x) maka f ‘(x) = nhn-1(x).h'(x)
f(x) = sin h(x) maka f ‘(x) = cos h(x).h'(x)
f(x) = cos h(x) maka f'(x) = -sin h(x).h'(x)
f(x) = tan h(x) maka f ‘(x) = sec2 h(x).h'(x)
f(x) = cot h(x) maka f ‘(x) = csc2 h(x).h'(x)
f(x) = sec h(x) maka f ‘(x) = sec h(x).tan h(x) h'(x)
f(x) = csc h(x) maka f ‘(x) = csc h(x).cot h(x) h'(x)
Contoh soal 1 :
f(x) = (x3 — 4x2 + 6x — 7)8 maka f ‘(x) = …
Jawab :
f ‘(x) = (x3 — 4x2 + 6x — 7)7 m(3x2 – 8x + 6)
Contoh soal 2 :
f(x) = sin9 x maka f ‘ (x) = …
Jawab :
f(x) = 9 sin8 x cos x
Contoh soal 3 :
f(x) = cos12 x maka f ‘ (x) = …
Jawab :
f(x) = 12 cos11 x (-sin x) = — 12cos11 x sin x
Contoh soal 4 :
f(x) = tan5 x maka f ‘ (x) = …
Jawab :
f(x) = 5 tan4 x sec2 x
Contoh soal 5 :
f(x) = cot6 x maka f ‘ (x) = …
Jawab :
f(x) = 6 cot5 x (-csc2 x) = – 6 cot5 x csc2 x
Contoh soal 6 :
f(x) = sec7 x maka f ‘ (x) = …
Jawab :
f(x) = 7 sec6 x (sec x tan x) = 7 sec7 x tan x
Contoh soal 7 :
f(x) = csc8 x maka f ‘ (x) = …
Jawab :
f(x) = 8 csc7 x (-csc x cot x) = – 8 csc8 x cot x
Contoh soal 8 :
f(x) = sin (3x2 — 5x) maka f ‘ (x) = …
Jawab :
f ‘(x) = cos (3x2 — 5x) . (6x — 5) = (6x — 5)cos (3x2 — 5x)
Contoh soal 9 :
f(x) = cos (x3 + 4x2 – 9x) maka f ‘ (x) = …
Jawab :
f ‘(x) = — sin (x3 + 4x2 – 9x) . (3x2 + 8x — 9)
f ‘(x) = — (3x2 + 8x — 9) sin (x3 + 4x2 – 9x)
Contoh soal 10 :
f(x) = tan (x2 + 9x) maka f ‘ (x) = …
Jawab :
f ‘(x) = sec2 (x2 + 9x) . (2x + 9)
f ‘(x) = (2x + 9) sec2 (x2 + 9x)
Contoh soal 11 :
f(x) = cot (7x — x2) maka f ‘ (x) = …
Jawab :
f ‘(x) = — csc2 (7x — x2) . (7 — 2x)
f ‘(x) = — (7 — 2x) csc2 (7x — x2)
f ‘(x) = (2x — 7) csc2 (7x — x2)
Contoh soal 12 :
f(x) = sec (x4– 3x) maka f ‘ (x) = …
Jawab :
f ‘(x) = sec (x4– 3x) tan (x4 – 3x) . (4x3 – 3)
f ‘(x) = (4x3 – 3) sec (x4– 3x) tan (x4 – 3x)
Contoh soal 13 :
f(x) = csc (x5– 7x2) maka f ‘ (x) = …
Jawab :
f ‘(x) = – csc (x5– 7x2) cot (x5– 7x2) (5x4– 14x)
f ‘(x) = – (5x4– 14x) csc (x5– 7x2) cot (x5– 7x2)
Contoh soal 14 :
f(x) = sin5 (7x2 – 6x) maka f ‘ (x) = ..
Jawab :
f ‘(x) = 5 sin4 (7x2 – 6x). cos (7x2 – 6x) . (14x — 6)
f ‘(x) = 5 (14x — 6) sin4 (7x2 – 6x). cos (7x2 – 6x)
f ‘(x) = 10 (7x — 3) sin4 (7x2 – 6x). cos (7x2 – 6x)
Contoh soal 15 :
f(x) = cos8 (x3 – 8x) maka f ‘ (x) = ..
Jawab :
f ‘(x) = 8 cos7 (x3 – 8x). [- sin (x3 – 8x)] . (3x2 — 8)
f ‘(x) = — 8 (3x2 — 8) cos7 (x3 – 8x). sin (x3 – 8x)
Contoh soal 16 :
f(x) = tan6 (2x4 – 3x + 7) maka f ‘ (x) = ..
Jawab :
f ‘(x) = 6 tan5 (2x4 – 3x + 7). sec2 (2x4 – 3x + 7) . (8x3 — 3)
f ‘(x) = 6 (8x3 — 3) tan5 (2x4 – 3x + 7). sec2 (2x4 – 3x + 7)
Contoh soal 17 :
f(x) = cot4 (2x2 + 6x — 1) maka f ‘ (x) = ..
Jawab :
f ‘(x) = 4 cot3 (2x2 + 6x — 1). [- csc2 (2x2 + 6x — 1)] . (4x + 6)
f ‘(x) = — 4(4x + 6) cot3 (2x2 + 6x — 1). csc2 (2x2 + 6x — 1)
f ‘(x) = — 8(2x + 3) cot3 (2x2 + 6x — 1). csc2 (2x2 + 6x — 1)
Contoh soal 18 :
f(x) = sec9 (x5 + 4x3) maka f ‘ (x) = ..
Jawab :
f ‘(x) = 9 sec8 (x5 + 4x3). sec (x5 + 4x3) . tan (x5 + 4x3) (5x4 + 12x2)
f ‘(x) = 9 (5x4 + 12x2) sec9 (x5 + 4x3). tan (x5 + 4x3)
Contoh soal 19 :
f(x) = csc12 (5x2 – 4x3) maka f ‘ (x) = ..
Jawab :
f ‘(x) = 12csc11 (5x2 – 4x3) .[- csc (5x2 – 4x3)] cot (5x2 – 4x3). (10x — 12x3)
f ‘(x) = — 12(10x — 12x3) csc12 (5x2 – 4x3) . cot (5x2 – 4x3)
f ‘(x) = — 24(5x — 6x3) csc12 (5x2 – 4x3) . cot (5x2 – 4x3)
f ‘(x) = 24(6x3 – 5x) csc12 (5x2 – 4x3) . cot (5x2 – 4x3)
Contoh soal 20 :
f(x) = sin16 x maka f ‘ (x) = ..
(A) cos16 x
(B) 16 sin15 x
(C) 16 cos15 x
(D) 8 sin14 x sin 2x
(E) 8 sin14 x cos 2x
Jawab : D
Ingat : 2 sin x cos x = sin 2x
f ‘(x) = 16 sin15 x cos x
f ‘(x) = 16 sin14 x sin x cos x
f ‘(x) = 8 sin14 x (2 sin x cos x)
f ‘(x) = 8 sin14 x sin 2x
Contoh soal 21 :
f(x) = cos9 4x maka f ‘ (x) = ..
(A) — sin9 4x
(B) – 4 sin9 4x
(C) 9 cos8 4x
(D) 36cos8 4x
(E) — 18cos7 4x sin 8x
Jawab : E
f ‘(x) = 9 cos8 4x . [- sin 4x] . 4
f ‘(x) = — 36 cos8 4x . sin 4x
f ‘(x) = — 36 cos7 4x . sin 4x cos 4x
f ‘(x) = — 18 cos7 4x . 2 sin 4x cos 4x
f ‘(x) = — 18 cos7 4x . sin 8x
Contoh soal 22 :
Jika f(x) = tan2 x dan g(x) = sec2 x buktikan bahwa f ‘ (x) = g'(x)
Jawab :
f(x) = tan2 x
f ‘ (x) = 2 tan x sec2 x
f’ (x) = 2 sec2 x tan x
g(x) = sec2 x
g’ (x) = 2sec x. sec x tan x
g’ (x) = 2 sec2 x tan x
Jadi, terbukti bahwa f ‘(x) = g ‘(x)