PENGERTIAN TURUNAN Turunan adalah suatu perhitungan terhadap perubahan nilai fungsi karena perubahan nilai input (variabel). Turunan dapat disebut juga sebagai diferensial dan proses dalam menentukan turunan suatu fungsi disebut sebagai diferensiasi.
SIFAT-SIFAT TURUNAN 1. Jika f(x)=c f ( x ) = c c c f'(x)=0 f ′ ( x ) = 0
Contoh:\begin{aligned} f(x)&=2 &\rightarrow f'(x)=0\\ f(x)&=13 &\rightarrow f'(x)=0\\ f(x)&=100 &\rightarrow f'(x)=0 \end{aligned} f ( x ) f ( x ) f ( x ) = 2 = 1 3 = 1 0 0 → f ′ ( x ) = 0 → f ′ ( x ) = 0 → f ′ ( x ) = 0
2. Jika f(x)=cx f ( x ) = c x f'(x)=c f ′ ( x ) = c
Contoh: \begin{aligned} f(x)&=2x &\rightarrow &f'(x)=2\\ f(x)&=13x &\rightarrow &f'(x)=13\\ f(x)&=100x &\rightarrow &f'(x)=100 \end{aligned} f ( x ) f ( x ) f ( x ) = 2 x = 1 3 x = 1 0 0 x → → → f ′ ( x ) = 2 f ′ ( x ) = 1 3 f ′ ( x ) = 1 0 0 3. Jika f(x)=x^n f ( x ) = x n f'(x)=nx^{n-1} f ′ ( x ) = n x n − 1
Contoh: \begin{aligned} f(x)&=x^4 &\rightarrow &f'(x)=4x^3\\ f(x)&=x^3 &\rightarrow &f'(x)=3x^2\\ f(x)&=x^2 &\rightarrow &f'(x)=2x \end{aligned} f ( x ) f ( x ) f ( x ) = x 4 = x 3 = x 2 → → → f ′ ( x ) = 4 x 3 f ′ ( x ) = 3 x 2 f ′ ( x ) = 2 x 4. Jika f(x)=cx^n f ( x ) = c x n f'(x)=cnx^{n-1} f ′ ( x ) = c n x n − 1
Contoh: \begin{aligned} f(x)&=2x^4 &\rightarrow &f'(x)=8x^3\\ f(x)&=13x^3 &\rightarrow &f'(x)=39x^2\\ f(x)&=100x^2 &\rightarrow &f'(x)=200x \end{aligned} f ( x ) f ( x ) f ( x ) = 2 x 4 = 1 3 x 3 = 1 0 0 x 2 → → → f ′ ( x ) = 8 x 3 f ′ ( x ) = 3 9 x 2 f ′ ( x ) = 2 0 0 x 5. Jika f(x)=c\,u(x) f ( x ) = c u ( x ) f'(x)=c\,u'(x) f ′ ( x ) = c u ′ ( x )
Contoh: \begin{aligned} f(x)&=4\ln{x}&\rightarrow &f'(x)=4\frac{1}{x}\\ f(x)&=3\cos{x}&\rightarrow &f'(x)=3\sin{x}\\ f(x)&=2\sin{x}&\rightarrow &f'(x)=-2\cos{x} \end{aligned} f ( x ) f ( x ) f ( x ) = 4 ln x = 3 cos x = 2 sin x → → → f ′ ( x ) = 4 x 1 f ′ ( x ) = 3 sin x f ′ ( x ) = − 2 cos x 6. Jika f(x)=u(x)\pm v(x) f ( x ) = u ( x ) ± v ( x ) f'(x)=u'(x)\pm v'(x) f ′ ( x ) = u ′ ( x ) ± v ′ ( x )
Contoh: \begin{aligned} f(x)&=2x+x^2&\rightarrow &f'(x)=2+2x\\ f(x)&=x^4-x^3&\rightarrow &f'(x)=4x^3-3x^2\\ f(x)&=\sin{x}+\cos{x}&\rightarrow &f'(x)=\cos{x}-\sin{x} \end{aligned} f ( x ) f ( x ) f ( x ) = 2 x + x 2 = x 4 − x 3 = sin x + cos x → → → f ′ ( x ) = 2 + 2 x f ′ ( x ) = 4 x 3 − 3 x 2 f ′ ( x ) = cos x − sin x 7. Jika f(x)=u(x)v(x) f ( x ) = u ( x ) v ( x ) f'(x)=u'(x)v(x)+u(x)v'(x) f ′ ( x ) = u ′ ( x ) v ( x ) + u ( x ) v ′ ( x )
Contoh: f(x)=x^4x^3 f ( x ) = x 4 x 3 Misalkan u(x)=x^4 u ( x ) = x 4 dan v(x)=x^3, v ( x ) = x 3 , maka u'(x)=4x^3 u ′ ( x ) = 4 x 3 dan v'(x)=3x^2, v ′ ( x ) = 3 x 2 , sehingga \begin{aligned} f'(x)&=(4x^3)(x^3)+(x^4)(3x^2)\\ &=4x^6+3x^6\\ &=7x^6 \end{aligned} f ′ ( x ) = ( 4 x 3 ) ( x 3 ) + ( x 4 ) ( 3 x 2 ) = 4 x 6 + 3 x 6 = 7 x 6 8. Jika f(x)=\displaystyle\frac{u(x)}{v(x)} f ( x ) = v ( x ) u ( x ) f'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{(v(x))^2} f ′ ( x ) = ( v ( x ) ) 2 u ′ ( x ) v ( x ) − u ( x ) v ′ ( x )
Contoh: f(x)=\frac{x^4}{x^3} f ( x ) = x 3 x 4 Misalkan u(x)=x^4 u ( x ) = x 4 dan v(x)=x^3, v ( x ) = x 3 , maka u'(x)=4x^3 u ′ ( x ) = 4 x 3 dan v'(x)=3x^2, v ′ ( x ) = 3 x 2 , sehingga \begin{aligned} f'(x)&=\frac{(4x^3)(x^3)-(x^4)(3x^2)}{(x^3)^2}\\ &=\frac{4x^6-3x^6}{x^6}\\ &=1 \end{aligned} f ′ ( x ) = ( x 3 ) 2 ( 4 x 3 ) ( x 3 ) − ( x 4 ) ( 3 x 2 ) = x 6 4 x 6 − 3 x 6 = 1 9. Jika f(x)={u(x)}^n f ( x ) = u ( x ) n f'(x)=n(u(x))^{n-1}u'(x) f ′ ( x ) = n ( u ( x ) ) n − 1 u ′ ( x )
Contoh: f(x)=(2x+x^2)^4 f ( x ) = ( 2 x + x 2 ) 4 Misalkan u(x)=2x+x^2, u ( x ) = 2 x + x 2 , sehingga u'(x)=2+2x, u ′ ( x ) = 2 + 2 x , maka f'(x)=4\left(2x+x^2\right)^3(2+2x) f ′ ( x ) = 4 ( 2 x + x 2 ) 3 ( 2 + 2 x ) CONTOH SOAL Soal 1.
Tentukan turunan dari f(x) = (2x + 1)4
Jawab:
Misalnya:
u(x) = 2x + 1 ⇒ u'(x) = 2 n = 4 n-1 . u'(x)4-1 . 23
Soal 2.
Tentukan turunan dari y = (x2 − 3x)7
Jawab :
y’ = 7(x2 − 3x)7-1 . (2x − 3)2 − 3x)6
Soal 3
Tentukanlah turunan fungsi dari f (x ) = 2x (x 4 – 5).
Jawab:
Misalkan jika u (x ) = 2x dan v (x ) = x 4 – 5, maka:
u ‘ (x ) = 2 dan v ‘ (x ) maka = 4x 3
Dengan begitu, akan didapatkan penjabaran serta hasilnya:
f ‘(x ) = u ‘(x ).v (x ) + u (x ).v ’(x ) = 2(x 4 – 5) + 2x (4x 3 ) = 2x 4 – 10 + 8x 4 = 10x 4 – 10
Soal 4. Soal Turunan Fungsi Aljabar
Turunan fungsi pertama dari
Jawab:
Soal ini merupakan soal fungsi yang berbentuk y = aun yang dapat dibahas dan diselesaikan dengan menggunakan rumus y’ = n . a . un-1 . Maka:
Sehingga turunannya adalah:
Soal 5. Turunan Fungsi Trigonometri
Tentukan turunan pertama dari:
Jawab:
Untuk menyelesaikan perosalan di atas, kita bisa memanfaatkan rumus campuran yakni:
serta juga bisa menggunakan rumus y’ = n. u’ sinn-1 u . cos u
Sehingga:
Soal 5.
Turunan dari f(x) = (x – 1)2 (2x + 3) adalah…
Jawab:
Misalkan:
u = (x − 1)2 ⇒ u’ = 2x − 2
f ‘(x) = u’v + uv’2 . 22 + 2x − 6 + 2(x2 − 2x + 1)2 + 2x − 6 + 2x2 − 4x + 22 − 2x − 4
Soal 6.
Apabila f(x) = x² – (1/x) + 1, maka f'(x) = . . . .
A. x – x²-2 + 12 – 1-2
Jawab:
f(x) = x2 – (1/x) + 1
= x2 – x-1 + 1
f'(x) = 2x -(-1)x-1-1
= 2x + x-2
Jawabannya: E
yaitu …
