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Calculating derivatives of various functions
1.
Nama : Siti
Fatimah NPM : 0031427 Kelas : 1 EA Mata Kuliah : Matematika2 Carilahnilai turunandari fungsi berikutini. 1 . 𝑦 = 𝑥7 + 6𝑥5 + 12𝑥3 2 . 𝑦 = 𝑥−5 + 2𝑥−3 + 3𝑥−7 3 . 𝑦 = √𝑥2 + √𝑥73 4 . 𝑦 = 2 √ 𝑥 + 6 √𝑥3 5 . 𝑦 = 𝑥2.sin 2𝑥 6 . 𝑦 = ( 𝑥2 + 1). cos3𝑥 7 . 𝑦 = sin 5𝑥.tan 2𝑥 8 . 𝑦 = cos3𝑥.cot4𝑥 9 . 𝑦 = sin5𝑥 cos7𝑥 10 . 𝑦 = cos 6𝑥 tan 2𝑥 Penyelesaian: 1 . 𝑦 = 𝑥7 + 6𝑥5 + 12𝑥3 𝑦′ = 7𝑥6 + 30𝑥4 + 36𝑥2 2 . 𝑦 = 𝑥−5 + 2𝑥−3 + 3𝑥−7 𝑦′ = −5𝑥−6 − 6𝑥−4 − 21𝑥−8 3 . 𝑦 = √𝑥2 + √𝑥73 𝑦 = 𝑥 + 𝑥 7 3 𝑦′ = 1 + 7 3 𝑥 7 3 −3
2.
𝑦′ = 1
+ 7 3 𝑥 4 3 𝑦′ = 1 + 7 3 √𝑥43 4 . 𝑦 = 2 √ 𝑥 + 6 √𝑥3 𝑦 = 2 𝑥 1 2 + 6 𝑥 3 2 𝑦 = 2𝑥− 1 2 + 6𝑥− 3 2 𝑦′ = −𝑥− 3 2 − 9𝑥− 5 2 𝑦′ = − 1 𝑥 3 2 − 9 𝑥 5 2 𝑦′ = − 1 √𝑥3 − 9 √ 𝑥5 5 . 𝑦 = 𝑥2.sin 2𝑥 Misal 𝑢 = 𝑥2 → 𝑢′ = 2𝑥 𝑣 = sin 2𝑥 → 𝑣′ = 2. cos 𝑥 𝑦′ = 𝑢′ 𝑣 + 𝑢𝑣′ 𝑦′ = 2𝑥. sin 2𝑥 + 𝑥2.2cos 𝑥 𝑦′ = 2𝑥. sin 2𝑥 + 2𝑥2.cos 𝑥 6 . 𝑦 = ( 𝑥2 + 1). cos3𝑥 Misal 𝑢 = 𝑥2 + 1 → 𝑢′ = 2𝑥 𝑣 = cos3𝑥 → 𝑣′ = −3.sin 3𝑥 𝑦′ = 𝑢′ 𝑣 + 𝑢𝑣′ 𝑦′ = 2𝑥. cos 3𝑥 + ( 𝑥2 + 1). −3.sin 3𝑥 𝑦′ = 2𝑥. cos3𝑥 + (−3𝑥2 − 3).sin 3𝑥 7 . 𝑦 = sin 5𝑥.tan 2𝑥 Misal 𝑢 = sin 5𝑥 → 𝑢′ = 5. cos5𝑥 𝑣 = tan 2𝑥 → 𝑣′ = 2. 𝑠𝑒𝑐2 2𝑥
3.
𝑦′ = 𝑢′
𝑣 + 𝑢𝑣′ 𝑦′ = 5. cos5𝑥.tan 2𝑥 + sin 5𝑥. 2. 𝑠𝑒𝑐2 2𝑥 𝑦′ = 5. cos5𝑥.tan 2𝑥 + 2. sin 5𝑥. 𝑠𝑒𝑐2 2𝑥 8 . 𝑦 = cos3𝑥.cot4𝑥 Misal 𝑢 = cos3𝑥 → 𝑢′ = −3. sin 3𝑥 𝑣 = cot4𝑥 → 𝑣′ = −4. 𝑐𝑜𝑠𝑒𝑐 2 4𝑥 𝑦′ = 𝑢′ 𝑣 + 𝑢𝑣′ 𝑦′ = −3. sin 3𝑥. cot4𝑥 + cos3𝑥 . − 4. 𝑐𝑜𝑠𝑒𝑐 2 4𝑥 𝑦′ = −3. sin 3𝑥 . cot4𝑥 + 4. cos3𝑥 . 𝑐𝑜𝑠𝑒𝑐 2 4𝑥 9 . 𝑦 = sin5𝑥 cos7𝑥 Misal 𝑢 = sin 5𝑥 → 𝑢′ = 5.cos5𝑥 𝑣 = cos7𝑥 → 𝑣′ = −7. sin 7𝑥 𝑦′ = 𝑢′. 𝑣 − 𝑢. 𝑣′ 𝑣2 𝑦′ = 5.cos5𝑥 .cos7𝑥 − sin 5𝑥 .−7.sin 7𝑥 (cos7𝑥)2 𝑦′ = 5.cos5𝑥 .cos7𝑥 + 7.sin 5𝑥.sin 7𝑥 𝑐𝑜𝑠27𝑥 10 . 𝑦 = cos 6𝑥 tan 2𝑥 Misal 𝑢 = cos6𝑥 → 𝑢′ = −6 .sin 6𝑥 𝑣 = tan2𝑥 → 𝑣′ = 2. 𝑠𝑒𝑐22𝑥 𝑦′ = 𝑢′. 𝑣 − 𝑢. 𝑣′ 𝑣2 𝑦′ = −6 .sin 6𝑥 . tan2𝑥 − cos6𝑥.2. 𝑠𝑒𝑐22𝑥 (tan 2𝑥)2 𝑦′ = −6 .sin 6𝑥 . tan2𝑥 − 2 .cos6𝑥. 𝑠𝑒𝑐22𝑥 𝑡𝑎𝑛2 2𝑥
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