15 Ejercicios de Trigonometricas

UNIVERSIDAD CENTRAL DEL ECUADOR Facultad de filosofía, letras y Ciencia de la Educación

Apellidos: Fuentes Almeida

Semestre: Tercero

Nombres: Dayanna Mishel

Paralelo: 3 “A”

Fecha: 30 de agosto del 2021

5 ejercicios de derivadas trigonométricas con producto Ejercicio 1 𝐲 = 𝐜𝐨𝐭 𝟐 (𝟒𝐱) ∗ 𝐭𝐚𝐧𝟑 (𝟒𝒙) y′ = 𝑐𝑜𝑡(4x) ∗ 3(tan2 (4x)sec2 (4x)(4) + tan3 (4𝑥 ) − csc2 (4𝑥)(4) y′ = 𝑐𝑜𝑡(4x)12(tan2 (4x))sec2 (4x) + tan3 (4𝑥 ) − 4 csc2 (4𝑥) y′ = 12 cot(4𝑥 ) tan2 (4x) sec2 (4x) − 4 tan3 (4𝑥 ) csc2 (4𝑥) Ejercicio 2 𝐲 = 𝐬𝐞𝐧𝟒 (𝟓𝐱) ∗ 𝐜𝐨𝐬𝟑 (𝟓𝒙) y′ = sen4 (5x) ∗ 3(cos 2 (5x)) − sen2 (5x)(5) + cos 3 (5𝑥 )4(sen3 (5𝑥)cos (5𝑥)(5) y′ = sen4 (5x) − 15(cos 2 (5x)sen(5x) + cos 3 (5𝑥 )20 sen3 (5𝑥)cos (5𝑥) y′ = −15 sen4 (5x) cos 2 (5x)𝑠𝑒𝑛(5𝑥) + 20 cos 3 (5𝑥 ) sen3 (5𝑥)cos (5𝑥)

Ejercicio 3 𝐲 = 𝐜𝐨𝐬𝟐 (𝟑𝐱) ∗ 𝐭𝐚𝐧𝟓 (𝟑𝒙) y′ = sen4 (5x) ∗ 3(cos 2 (5x)) − sen2 (5x)(5) + cos 3 (5𝑥 )4(sen3 (5𝑥)cos (5𝑥)(5) y′ = sen4 (5x) − 15(cos 2 (5x)sen(5x) + cos 3 (5𝑥 )20 sen3 (5𝑥)cos (5𝑥)

y′ = −15 sen4 (5x) cos 2 (5x)𝑠𝑒𝑛(5𝑥) + 20 cos 3 (5𝑥 ) sen3 (5𝑥)cos (5𝑥)

Ejercicio 4 𝐲 = 𝐬𝐢𝐧𝟐 (𝟐𝐱) ∗ 𝐜𝐨𝐬(𝟐𝒙) y′ = 2sen(2x) cos(2x) 2 cos(2x) + sen2 (2x)(−sin (2𝑥 )(2) y′ = 2sen(2x) cos(2x) 2 cos(2x) + sen2 (2x)(−sin (2𝑥 )(2) y′ = 4 sin(2𝑥 ) cos 2 (2𝑥 ) − 2 cos 3 (2𝑥 ) Ejercicio 5 𝐲 = 𝐜𝐨𝐬𝟐 (𝟔𝐱) ∗ 𝐭𝐚𝐧𝟑 (𝟔𝒙) 3sen2 (6x)𝑐𝑜𝑠(6x)6𝑐𝑜𝑠(6𝑥 ) − sin3 (6𝑥 )(−sen3 (6𝑥)(6) y = cos 2 (6x) ′

18sen2 (6x)cos 2 (6x) − sen3 (6x)(−6 sen3 )(6𝑥) y = cos 2 (6x) ′

y′ =

3sen2 (12x) + sen2 (6x) cos 2 (6x)

5 ejercicios de derivadas trigonométricas de división

Ejercicio 1 𝐲=

𝒄𝒐𝒔 (𝟑𝒙) 𝒔𝒆𝒏 (𝟓𝒙)

𝑦′ =

𝑠𝑒𝑛 (5𝑥 )(−3 𝑠𝑒𝑛 (3𝑥 )) − cos(3𝑥)(5 cos (5𝑥)) sen2 (5𝑥)

𝑦′ =

−3 𝑠𝑒𝑛 (3𝑥 ) 𝑠𝑒𝑛 (5𝑥 ) − 5 cos(3𝑥 ) cos(5𝑥) sen2 (5𝑥)

𝑦′ = −

3 𝑠𝑒𝑛 (3𝑥 ) 𝑠𝑒𝑛 (5𝑥 ) + 5 cos(3𝑥 ) cos(5𝑥) sen2 (5𝑥)

Ejercicio 2 𝐲=

𝐬𝐞𝐧𝟐 (𝟔𝒙) 𝐜𝐨𝐬𝟐 (𝟕𝒙)

𝑦′ =

cos 2 (7𝑥 ) 2 𝑠𝑒𝑛(6𝑥 )𝑐𝑜𝑠(6𝑥 )(6) − sen2 (6𝑥 ) 2𝑐𝑜𝑠(7𝑥 ) − (𝑠𝑒𝑛(7𝑥 )(7)) cos 4 (7𝑥)

2 𝑠𝑒𝑛(6𝑥 )𝑐𝑜𝑠(6𝑥 ) 6 cos 2 (7𝑥 ) − sen2 (6𝑥 ) 2𝑐𝑜𝑠(7𝑥 ) − (𝑠𝑒𝑛(7𝑥 )(7)) 𝑦 = sen2 (5𝑥) ′

𝑦′ =

6 𝑠𝑒𝑛 (12𝑥 ) 𝑐𝑜𝑠 (7𝑥 ) + 14 sen(6x)2 𝑠𝑒𝑛 (7𝑥) cos 3 (7𝑥)

Ejercicio 3 𝐲=

𝒔𝒆𝒏 (𝟓𝒙) 𝒄𝒐𝒔 (𝟓𝒙)

𝑦′ =

𝑐𝑜𝑠 (5𝑥 )5 cos(5𝑥 ) − 𝑠𝑒𝑛(5𝑥 )(−5𝑠𝑒𝑛(5𝑥 )) cos 2 (5𝑥)

5 cos 2 (5𝑥 ) + 5 sen2 (5𝑥) 𝑦 = cos 2 (5𝑥) ′

𝑦′ =

𝑦′ =

5(cos 2 (5𝑥 ) + sen2 (5𝑥 )) 𝑐𝑜𝑠 2(5𝑥) 5 𝑐𝑜𝑠 2 (5𝑥)

𝑦 ′ = 5 𝑠𝑒𝑐 2(5𝑥) Ejercicio 4 𝐲=

𝒔𝒆𝒏 (𝟖𝒙) 𝒄𝒐𝒔 (𝟖𝒙)

𝑦′ =

𝑐𝑜𝑠 (8𝑥 )8 cos(5𝑥 ) − 𝑠𝑒𝑛(8𝑥 )(−5𝑠𝑒𝑛(8𝑥 )) cos 2 (8𝑥)

8 cos 2 (8𝑥 ) + 8 sen2 (8𝑥) 𝑦 = cos 2 (8𝑥) ′

8(cos 2 (8𝑥 ) + sen2 (8𝑥 )) 𝑦 = 𝑐𝑜𝑠 2(8𝑥) ′

𝑦′ =

8 𝑐𝑜𝑠 2 (8𝑥)

𝑦 ′ = 8 𝑠𝑒𝑐 2(8𝑥)

Ejercicio 5 𝐲=

𝒔𝒆𝒏 (𝟏𝟏𝒙) 𝒄𝒐𝒔 (𝟏𝟏𝒙)

𝑦′ =

𝑐𝑜𝑠 (11𝑥 )11 cos(11𝑥 ) − 𝑠𝑒𝑛(11𝑥 )(−11𝑠𝑒𝑛(11𝑥 )) cos 2 (11𝑥)

11 cos 2 (11𝑥 ) + 11 sen2 (11𝑥) 𝑦 = cos 2 (11𝑥) ′

𝑦′ = 𝑦′ =

11(cos 2 (11𝑥 ) + sen2 (11𝑥 )) 𝑐𝑜𝑠 2 (11𝑥) 11 𝑐𝑜𝑠 2 (11𝑥)

𝑦 ′ = 11 𝑠𝑒𝑐 2 (11𝑥)

5 ejercicios de derivadas trigonométricas de grado mayor que dos

Ejercicio 1 𝐲 = 𝐬𝐞𝐧𝟒 (𝟐𝐱)

y′ = 4 sen3 (2x) Dx sen (2x) y′ = 4 sen3 (2x) cos (2x)(2) y′ = 8 sen3 (2x) cos (2x) Ejercicio 2 𝐲 = 𝒄𝒐𝒔𝟒 (𝟐𝟓𝐱) y′ = 4 cos 3 (25x) Dx − sen (25x) y′ = 4 cos 3 (25x) − sen (25x)(25) y′ = −100 cos 3 (25x) sen (25x) Ejercicio 3 𝐲 = 𝐬𝐞𝐧𝟒 (𝟒𝐱) y′ = 4 sen3 (4x) Dx sen (4x) y′ = 4 sen3 (4x) cos (4x)(4) y′ = 16sen3 (4x) cos (4x) Ejercicio 4 𝐲 = 𝒄𝒐𝒔𝟒 (𝟏𝟔𝐱) y′ = 4 cos 3 (16x) Dx − sen (16x) y′ = 4 cos 3 (16x) − sen (16x)(16) y′ = −64 cos 3 (16x) sen (16x) Ejercicio 5 𝐲 = 𝐬𝐞𝐧𝟒 (𝟏𝟐𝐱) y′ = 4 sen3 (12x) Dx sen (12x) y′ = 4 sen3 (12x) cos (12x)(12) y′ = 48 sen3 (12x) cos (12x)