Turunan pertama dari fungsi hiperbol trigonometri berikut! y = x sinh x dan y = x^2/cosh⁡ x

Tentukanlah Turunan pertama dari fungsi hiperbol trigonometri berikut!

1. y = x sinh x

2. y = x^2/cosh⁡ x

Jawab:

1. dy/dx=1(sinh⁡ x) – x d/dx(sinh⁡ x)

dy/dx = sinh⁡ x + x cosh ⁡x

Atau

dy/dx = (e^x- e^(-x) )/2⁡+ x ((e^x+ e^(-x) )/2)

2. dy/dx =  (2x(cosh⁡ x) –  x^2 (sinh⁡ x)) / (cosh x)^2

Atau :

dy/dx =  (2x((e^x+ e^(-x) )/2)-x^2 ((e^x- e^(-x) )/2) )/((e^x+ e^(-x) )/2)^2

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Tentukanlah Turunan pertama dari fungsi hiperbol  trigonometri berikut! 1.y = x sinh x 2.y = x^2/cosh⁡ x

Rumus Turunan Pertama dari fungsi Invers Trigonometri dan fungsi hiperbol Trigonometri

Rumus Turunan Pertama dari fungsi Invers Trigonometri

1. d/dx(arc sin⁡ u )= 1/√(1- u^2 ) (du/dx)

2. d/dx(arc cos⁡ u )= (-1)/√(1- u^2 ) (du/dx)

3. d/dx ( arc tg u)= 1/(1+ u^2 ) (du/dx)

4. d/dx ( arc sec u)= 1/(|u| √(u^2- 1)) (du/dx)

5. d/dx ( arc ctg u)= (-1)/(1+ u^2 ) (du/dx)

6. d/dx ( arc cosec u)= (-1)/(|u| √(u^2- 1)) (du/dx)

Fungsi Hiperbol Trogonometri

1. Sinh = (e^x- e^(-x) )/2 cosh = (e^x+ e^(-x) )/2

2. tanh = (e^x- e^(-x) )/(e^x+ e^(-x) )

3. coth = (e^x+ e^(-x) )/(e^x- e^(-x) )

4. sech = (2 )/(e^x+ e^(-x) )

1. d/dx(sinh⁡ u )  = -coshu (du/dx) =  (e^x+ e^(-x) )/2 (du/dx)

2. d/dx(cosh⁡ u ) =  sinhu (du/dx) =  (e^x- e^(-x) )/2 (du/dx)

3. d/dx(tanh⁡ u ) =  sech^2 u (du/dx) = ((2 )/(e^x+ e^(-x) ))^2 (du/dx)

4. d/dx(coth⁡ u ) =Rumus Turunan Pertama dari fungsi hiperbol Trigonometri  – cosech^2 u (du/dx) = -((2 )/(e^x- e^(-x) ))^2 (du/dx)

5. d/dx(sech⁡ u ) = – sech u tanh⁡ u (du/dx) =  – ((2 )/(e^x+ e^(-x) ))^2 ((e^x- e^(-x) )/(e^x+ e^(-x) ))(du/dx)

6. d/dx(cosech⁡ u ) = – cosech u coth⁡ u (du/dx) =  – ((2 )/(e^x- e^(-x) ))^2 ((e^x+ e^(-x) )/(e^x- e^(-x) ))(du/dx)

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Rumus Turunan Pertama dari fungsi Invers Trigonometri , Fungsi Hiperbol Trogonometri , Rumus Turunan Pertama dari fungsi hiperbol Trigonometri

y = arc tg (x – 1)/(x +1)

Tentukan turunan pertama dari :

1. y = arc tg (x – 1)/(x +1) !

Jawab:

Karena: d/dx ( arc tg u)= 1/(1+ u^2 ) (du/dx)

d/dx (arc tg (x – 1)/(x +1))= 1/(1+ ((x – 1)/(x +1))^2 ) (d((x – 1)/(x +1))/dx)

d/dx (arc tg (x – 1)/(x +1))= 1/(((x+1)^2+ (x-1)^2)/(x+1)^2 ) [(1(x+1)- (x-1)1)/(x+1)^2 ]

d/dx (arc tg (x – 1)/(x +1))= (x+1)^2/((x+1)^2+ (x-1)^2 ) (2/(x+1)^2 )

d/dx (arc tg (x – 1)/(x +1))= 2/((x+1)^2+ (x-1)^2 )

d/dx (arc tg (x – 1)/(x +1))= 2/(( x^2+ 2x+1)+ (x^2- 2x+1))

d/dx (arc tg (x – 1)/(x +1))= 2/(2x^2+ 2)

d/dx (arc tg (x – 1)/(x +1))= 2/(2(x^2+ 1))

d/dx (arc tg (x – 1)/(x +1))= 1/(x^2+ 1)

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Tentukan turunan pertama dari y = arc tg (x – 1)/(x +1)

y = arc ctg 2/x+ arc tg x/2

Tentukanlah turunan pertama dari y = arc ctg 2/x+ arc tg x/2 !

Jawab:

Karena:

d/dx ( arc ctg u)= (- 1)/(1+ u^2 ) (du/dx)

Maka :

d/dx ( arc ctg 2/x)= (- 1)/(1+ (2/x)^2 ) (d(2/x)/dx)

d/dx ( arc ctg 2/x)= (- 1)/(1+ 4/x^2 ) ((-2)/x^2 )

d/dx ( arc ctg 2/x)= (-1)/((x^2+4)/x^2 ) ((-2)/x^2 )

d/dx ( arc ctg 2/x)=((-x^2)/(x^2+ 4)) ((-2)/x^2 )

d/dx ( arc ctg 2/x)=(2/(x^2+ 4))

Karena :

d/dx ( arc tg u) = 1/(1+ u^2 ) (du/dx)

Maka:

d/dx ( arc tg x/2)= 1/(1+ (x/2)^2 ) (d(x/2)/dx)

d/dx ( arc tg x/2)= 1/(1+ x^2/4) (2/4)

d/dx ( arc tg x/2)=(4/(4+ x^2 )) (2/4)

d/dx ( arc tg x/2)=(2/(4+ x^2 ))

Maka turunan dari y = arc ctg 2/x+ arc tg x/2 adalah

dy/dx= d/dx ( arc ctg 2/x)+ d/dx ( arc tg x/2)

dy/dx= (2/(x^2+ 4))+ (2/(4+ x^2 ))

dy/dx= ((2+2)/(x^2+ 4))

dy/dx= (4/(x^2+ 4))

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Tentukanlah turunan pertama dari y = arc ctg 2/x+ arc tg x/2

y = x^4 sin⁡ (1/x^2)

Tentukan turunan pertama dari y = x^4  sin⁡ (1/x^2)  !

Jawab:

y = uv maka : y^’= u^’ v+uv^’

Pertama cari dy/dx dari sin⁡ (1/x^2)

Maka:

dy/dx =  cos⁡ (1/x^2)  d(1/x^2 )/dx

dy/dx = – 2x/x^4 (cos 1/x^2 )

Kemudian carilah dy/dx dari y = x^4 sin⁡ (1/x^2 )

dy/dx =  d(x^4)/dx  sin⁡ (1/x^2 ) + x^4  d(sin⁡(1/x^2 )/dx

dy/dx = 4x^3 sin⁡ (1/x^2) + x^4 [- 2x/x^4 (cos 1/x^2 )]

dy/dx = 4x^3 sin⁡ (1/x^2 ) – 2x cos 1/x^2

dy/dx = 2x(2x^2 sin (1/x^2 ) – cos (1/x^2 )

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Tentukan turunan pertama dari y = x^4 sin⁡(1/x^2)