Turunan kedua dari x^4+ y^4 = 81

Carilah y^” dari fungsi x^4+ y^4 = 81 !

Jawab :

Gunakan metode turunan implisit

(d(x^4))/dx + (d(y^4))/dx= ((d(81))/dx

4x^3+(d(y^4 )/dy)(dy/dx )=0

4x^3+ 4y^3 y^’=0

y^’= -x^3/y^3

Maka :

y^”= -((d(x^3 )/dx)(y^3 )-x^3 (d(y^3 )/dx) )/(y^3)^2

y^”= -(3x^2 (y^3 )-x^3 (3y^2 )(y^’))/y^6

y^”= -(3x^2 y^3-x^3 (3y^2 )(-x^3/y^3 ))/y^6

y^”= -(3x^2 y^3+ 3x^6/y)/y^6

y^”=(-(3x^2 y^3+ 3x^6/y)/y^6 )(y/y)

y^”= -(3(x^2 y^4+ x^6) )/y^7

y^”= -(3x^2 (y^4+ x^4) )/y^7

Atau

y^”= -(3x^2 (81) )/y^7 = -243(x^2/y^7 )

for more detailed writing click on the following link Carilah y^” dari fungsi x^4+ y^4 = 81 !

sin (x + y) = y^2 cos⁡ x

Carilah dy/dx dari fungsi sin(x + y) = y^2 cos⁡x !

Jawab :

Gunakan metode turunan implisit

(d (sin(x + y) ))/dx = (d(y^2 cos⁡x ))/dx

cos (x + y) ((d(x+ y))/dx) = ((d(y^2))/dx)(cos⁡x)+(y^2 )((d(cos⁡x))/dx)

cos (x + y) [(d(x)/dx)+(dy/dx)]=(d(y^2 )/dy)(dy/dx)(cos⁡x)+ y^2 (- sin x)

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

cos (x + y)+ dy/dx cos (x+y) =2y dy/dx(cos⁡x)- y^2 (sin x)

cos (x + y) + y^2 sin⁡x = 2y dy/dx cos⁡x – dy/dx cos (x+y)

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

dy/dx= (cos (x + y) + y^2 sin⁡x)/(2y cos⁡x-cos ⁡(x+y) )

for more detailed writing click on the following link Carilah dy/dx dari fungsi sin(x + y) = y^2 cos⁡x !

(a) dy/dx dari fungsi x^2 + y^2=25 (b) Tentukan pula persamaan garis singgungnya di titik (3,4)

(a) Carilah dy/dx dari fungsi x^2+ y^2=25

(b) Tentukan persamaan garis singgungnya di titik (3,4)

Jawab:

(a) Gunakan metode turunan implisit

x^2+ y^2=25

maka∶ y= √(2&25- x^2 )

(d ( x^2+ y^2))/dx = (d( 25))/dx

(d( x^2))/dx+ (d( y^2))/dx = (d( 25))/dx

(d( x^2))/dx+ (d( y^2 )/dy)(dy/dx) = (d( 25))/dx

2x + (2y) ( dy/dx) =0

dy/dx= (-2x)/2y

dy/dx= – x/y

maka: dy/dx= – x/√(2&25- x^2 )

(b) Maka : m = – 3/√(2&25- 3^2 ) = – 3/√(2&16) = – 3/4

Maka persamaan garis singgung kurva dititik (3,4) adalah :

y – y_1=m( x- x_1)

y – 4 = – 3/4 ( x-3)

y = – 3/4 x+ 9/4+ 4

y = – 3/4 x + 25/4

atau 4y = – 3x + 25

atau 3x + 4y = 25

for more detailed writing click on the following link Carilah dy/dx dari fungsi x^2+ y^2=25 Tentukan persamaan garis singgungnya di titik (3,4)

(a) f(x) = (x^4+ 3x^2 – 2)^50 (b) f(x) = (1+ x^4 )^2/3

Tentukanlah turunan pertama dari setiap fungsi berikut:

(a) f(x) = (x^4+ 3x^2-2)^50

(b) f(x) = (1+ x^4 )^(2/3)

Jawab

(d [ f(x) ]^n)/dx=(n[f(x) ]^(n-1) ) ( f^’ (x))

Maka :

(a).

(d [x^4+ 3x^2-2)^50])/dx=[50(x^4+ 3x^2-2)^49] ( 4x^3+ 6x) = 50( 4x^3+ 6x) (x^4+ 3x^2-2)^49

(b).

(d [(1+ x^4 )^(2/3)] )/dx=[2/3(1+ x^4 )^(-1/3) ] ( 4x^3) = ( 2/(3(1+ x^4 )^(1/3) ) ) ( 4x^3) = (8x^3)/(3(1+ x^4 )^(1/3) ) = (8x^3)/(3√(3&1+ x^4 ))

for more detailed writing click on the following link Tentukanlah turunan pertama dari setiap fungsi berikut: f(x) = (x^4+ 3x^2-2)^50 f(x) = 〖(1+ x^4 )〗^(2/3)

f(θ) = (e )^(sec 3θ)

Tentukanlah turunan pertama : f(θ) = (e )^(sec 3θ) !

Jawab :

d (e)^f(θ)/dx= e^f(θ) d[f(θ) ]/dx

Maka :

d [(e )^(sec 3θ)]/dx= e^(sec 3θ) d( sec 3θ)/dx = e^(sec 3θ) 3 sec 3θ tan 3θ = 3e^(sec 3θ) sec 3θ tan 3θ

for more detailed writing click on the following link Tentukanlah turunan pertamanya : f(θ) = (e )^(sec 3θ) !