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Dengan menggunakan aturan rantai, tentukan turunan pertama setiap fungsi berikut. b. F ( x ) = ( 3 x 2 + 5 ) 5 ( 4 x − 1 ) 3 ⋅ ( x 2 + 2 ) 4

Dengan menggunakan aturan rantai, tentukan turunan pertama setiap fungsi berikut.

b. $F (x) = \frac{( 4 x - 1 ) ^{3} \cdot ( x ^{2} + 2 ) ^{4}}{( 3 x ^{2} + 5 ) ^{5}}$

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Ingat aturan turunan untuk: Perkalian fungsi f ( x ) = u ( x ) ⋅ v ( x ) adalah f ′ ( x ) = u ′ ( x ) ⋅ v ( x ) + u ( x ) ⋅ v ′ ( x ) . Pembagian fungsi f ( x ) = v u adalah f ′ ( x ) = ( v ( x )) 2 u ′ ( x ) ⋅ v ( x ) − u ( x ) ⋅ v ′ ( x ) Aturan rantai bagi fungsi f ( x ) = a ⋅ [ u ( x ) ] n adalah f ′ ( x ) = n ⋅ a ⋅ [ u ( x ) ] n ⋅ u ′ ( x ) Diketahui fungsi . Misalkan g ( x ) = ( 4 x − 1 ) 3 , h ( x ) = ( x 2 + 2 ) 4 , dan m ( x ) = ( 3 x 2 + 5 ) 5 , maka fungsi F ( x ) dapat dituliskan menjadi F ( x ) = m ( x ) g ( x ) ⋅ h ( x ) = m ( x ) N ( x ) Pertama kita cariturunan pertama dari masing-masing fungsi g ( x ) , h ( x ) , dan m ( x ) dengan menggunakan aturan rantai: turunan dari g ( x ) = ( 4 x − 1 ) 3 adalah g ′ ( x ) = = 3 ⋅ ( 4 x − 1 ) 3 − 1 ⋅ ( 4 ) 12 ( 4 x − 1 ) 2 turunan dari h ( x ) = ( x 2 + 2 ) 4 adalah h ′ ( x ) = = 4 ⋅ ( x 2 + 2 ) 4 − 1 ⋅ ( 2 x ) 8 x ( x 2 + 2 ) 3 turunan dari m ( x ) = ( 3 x 2 + 5 ) 5 adalah m ′ ( x ) = = 5 ⋅ ( 3 x 2 + 5 ) 5 − 1 ⋅ ( 6 x ) 30 x ( 3 x 2 + 5 ) 4 Selanjutnya kita akan menentukan turunan pertama dari fungsi N ( x ) = g ( x ) ⋅ h ( x ) menggunakan aturan turunan untuk perkalian fungsi: N ′ ( x ) = = = = = = g ′ ( x ) ⋅ h ( x ) + g ( x ) ⋅ h ′ ( x ) ( 12 ( 4 x − 1 ) 2 ) ⋅ ( x 2 + 2 ) 4 + ( 4 x − 1 ) 3 ⋅ ( 8 x ( x 2 + 2 ) 3 ) [ ( 4 ( 4x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ) ⋅ ( 3 ( x 2 + 2 ) ) ] + [ ( 4 ( 4x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ) ⋅ ( ( 4 x − 1 ) ⋅ 2 x ) ] 4 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ [ 3 ( x 2 + 2 ) + ( 4 x − 1 ) ⋅ 2 x ] 4 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ [ 3 x 2 + 6 + 8 x 2 − 2 x ] 4 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ [ 11 x 2 − 2 x + 6 ] Turunan untuk pembagian fungsi. F ′ ( x ) = ( m ( x )) 2 N ′ ( x ) ⋅ m ( x ) − N ( x ) ⋅ m ′ ( x ) = = = = = = N ′ ( x ) ⋅ m ( x ) − N ( x ) ⋅ m ′ ( x ) [ [ 4 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 11 x 2 − 2 x + 6 ) ] ⋅ [ 3 x 2 + 5 ] 5 ] − [ ( 4 x − 1 ) 3 ⋅ ( x 2 + 2 ) 4 ⋅ [ ( 30 x ) ⋅ ( 3 x 2 + 5 ) 4 ] ] [ ( 2 ( 4x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ) ⋅ ( 2 ( 11 x 2 − 2 x + 6 ) ⋅ ( 3 x 2 + 5 ) ) ] − [ ( 2 ( 4x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ) ⋅ ( ( 4 x − 1 ) ⋅ ( x 2 + 2 ) ⋅ 15 x ) ] [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ] ⋅ ( 2 ( 11 x 2 − 2 x + 6 ) ⋅ ( 3 x 2 + 5 ) − ( 4 x − 1 ) ⋅ ( x 2 + 2 ) ⋅ 15 x ] [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ] ⋅ [ ( 6 x 2 + 10 ) ( 11 x 2 − 2 x + 6 ) − ( 60 x 2 − 15 x ) ( x 2 + 2 ) ] [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ] ⋅ [ 66 x 4 − 12 x 3 + 36 x 2 + 110 x 2 − 20 x + 60 − ( 60 x 4 + 120 x 2 − 15 x 3 − 30 x ) ] [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ] ⋅ [ 6 x 4 + 3 x 3 + 26 x 2 + 10 x + 60 ] Dengan demikian, turunan pertama dari fungsi F ( x ) yaitu F ′ ( x ) = = = = ( m ( x )) 2 N ′ ( x ) ⋅ m ( x ) − N ( x ) ⋅ m ′ ( x ) ( ( 3 x 2 + 5 ) 5 ) 2 [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ] ⋅ [ 6 x 4 + 3 x 3 + 26 x 2 + 10 x + 60 ] ( 3 x 2 + 5 ) 10 [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ⋅ ( 3 x 2 + 5 ) 4 ] ⋅ [ 6 x 4 + 3 x 3 + 26 x 2 + 10 x + 60 ] ( 3 x 2 + 5 ) 6 [ 2 ( 4 x − 1 ) 2 ⋅ ( x 2 + 2 ) 3 ] ⋅ ⋅ [ 6 x 4 + 3 x 3 + 26 x 2 + 10 x + 60 ]

Ingat aturan turunan untuk:

Perkalian fungsi $f (x) = u (x) \cdot v (x)$ adalah

$f^{'} (x) = u^{'} (x) \cdot v (x) + u (x) \cdot v^{'} (x)$ .

Pembagian fungsi $f (x) = \frac{u}{v}$ adalah

$f^{'} (x) = \frac{u ^{'} ( x ) \cdot v ( x ) - u ( x ) \cdot v ^{'} ( x )}{( v ( x )) ^{2}}$

Aturan rantai bagi fungsi $f (x) = a \cdot [u (x)]^{n}$ adalah

$f^{'} (x) = n \cdot a \cdot [u (x)]^{n} \cdot u^{'} (x)$

Diketahui fungsi $begin mathsize 14px style F open parentheses x close parentheses equals fraction numerator open parentheses 4 x minus 1 close parentheses cubed times open parentheses x squared plus 2 close parentheses to the power of 4 over denominator open parentheses 3 x squared plus 5 close parentheses to the power of 5 end fraction end style$ .

Misalkan $g (x) = (4 x - 1)^{3}$ , $h (x) = (x^{2} + 2)^{4}$ , dan $m (x) = (3 x^{2} + 5)^{5}$ , maka fungsi $F (x)$ dapat dituliskan menjadi

$F (x) = \frac{g ( x ) \cdot h ( x )}{m ( x )} = \frac{N ( x )}{m ( x )}$

Pertama kita cari turunan pertama dari masing-masing fungsi $g (x), h (x), dan m (x)$ dengan menggunakan aturan rantai:

turunan dari $g (x) = (4 x - 1)^{3}$ adalah

$g^{'} (x) = = 3 \cdot (4 x - 1)^{3 - 1} \cdot (4) 12 (4 x - 1)^{2}$

turunan dari $h (x) = (x^{2} + 2)^{4}$ adalah

$h^{'} (x) = = 4 \cdot (x^{2} + 2)^{4 - 1} \cdot (2 x) 8 x (x^{2} + 2)^{3}$

turunan dari $m (x) = (3 x^{2} + 5)^{5}$ adalah

$m^{'} (x) = = 5 \cdot (3 x^{2} + 5)^{5 - 1} \cdot (6 x) 30 x (3 x^{2} + 5)^{4}$

Selanjutnya kita akan menentukan turunan pertama dari fungsi $N (x) = g (x) \cdot h (x)$ menggunakan aturan turunan untuk perkalian fungsi:

$N^{'} (x) = = = = = = g^{'} (x) \cdot h (x) + g (x) \cdot h^{'} (x) (12 (4 x - 1)^{2}) \cdot (x^{2} + 2)^{4} + (4 x - 1)^{3} \cdot (8 x (x^{2} + 2)^{3}) [(4 (4x - 1)^{2} \cdot (x^{2} + 2)^{3}) \cdot (3 (x^{2} + 2))] + [(4 (4x - 1)^{2} \cdot (x^{2} + 2)^{3}) \cdot ((4 x - 1) \cdot 2 x)] 4 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot [3 (x^{2} + 2) + (4 x - 1) \cdot 2 x] 4 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot [3 x^{2} + 6 + 8 x^{2} - 2 x] 4 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot [11 x^{2} - 2 x + 6]$

Turunan untuk pembagian fungsi.

$F^{'} (x) = \frac{N ^{'} ( x ) \cdot m ( x ) - N ( x ) \cdot m ^{'} ( x )}{( m ( x )) ^{2}}$

$= = = = = = N^{'} (x) \cdot m (x) - N (x) \cdot m^{'} (x) [[4 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (11 x^{2} - 2 x + 6)] \cdot [3 x^{2} + 5]^{5}] - [(4 x - 1)^{3} \cdot (x^{2} + 2)^{4} \cdot [(30 x) \cdot (3 x^{2} + 5)^{4}]] [(2 (4x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (3 x^{2} + 5)^{4}) \cdot (2 (11 x^{2} - 2 x + 6) \cdot (3 x^{2} + 5))] - [(2 (4x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (3 x^{2} + 5)^{4}) \cdot ((4 x - 1) \cdot (x^{2} + 2) \cdot 15 x)] [2 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (3 x^{2} + 5)^{4}] \cdot (2 (11 x^{2} - 2 x + 6) \cdot (3 x^{2} + 5) - (4 x - 1) \cdot (x^{2} + 2) \cdot 15 x] [2 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (3 x^{2} + 5)^{4}] \cdot [(6 x^{2} + 10) (11 x^{2} - 2 x + 6) - (60 x^{2} - 15 x) (x^{2} + 2)] [2 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (3 x^{2} + 5)^{4}] \cdot [66 x^{4} - 12 x^{3} + 36 x^{2} + 110 x^{2} - 20 x + 60 - (60 x^{4} + 120 x^{2} - 15 x^{3} - 30 x)] [2 (4 x - 1)^{2} \cdot (x^{2} + 2)^{3} \cdot (3 x^{2} + 5)^{4}] \cdot [6 x^{4} + 3 x^{3} + 26 x^{2} + 10 x + 60]$

Dengan demikian, turunan pertama dari fungsi $F (x)$ yaitu

$F^{'} (x) = = = = \frac{N ^{'} ( x ) \cdot m ( x ) - N ( x ) \cdot m ^{'} ( x )}{( m ( x )) ^{2}} \frac{[ 2 ( 4 x - 1 ) ^{2} \cdot ( x ^{2} + 2 ) ^{3} \cdot ( 3 x ^{2} + 5 ) ^{4} ] \cdot [ 6 x ^{4} + 3 x ^{3} + 26 x ^{2} + 10 x + 60 ]}{( ( 3 x ^{2} + 5 ) ^{5} ) ^{2}} \frac{[ 2 ( 4 x - 1 ) ^{2} \cdot ( x ^{2} + 2 ) ^{3} \cdot ( 3 x ^{2} + 5 ) ^{4} ] \cdot [ 6 x ^{4} + 3 x ^{3} + 26 x ^{2} + 10 x + 60 ]}{( 3 x ^{2} + 5 ) ^{10}} \frac{[ 2 ( 4 x - 1 ) ^{2} \cdot ( x ^{2} + 2 ) ^{3} ] \cdot \cdot [ 6 x ^{4} + 3 x ^{3} + 26 x ^{2} + 10 x + 60 ]}{( 3 x ^{2} + 5 ) ^{6}}$