Editorial - Matlan - Turunan Fungsi Trigonometri

Problem 1

(y^′)=sin(3/x)⋅3/(x2)=3/(x2)*sin(3/x)

(C)

Problem 2

(g^′)(x)=3*cos(x)+sin(x)

(B)

Problem 3

(y^′)=-sin(2*x3-x2)⋅(6*x2-2*x)

(E)

Problem 4

y=x2*sin(3*x)

u=x2

(u^′)=2*x

v=sin(3*x)

(v^′)=3*cos(3*x)

(y^′)=2*x⋅sin(3*x)+x2⋅3*cos(3*x)=2*x*sin(3*x)+3*x2*cos(3*x)

(B)

Problem 5

(ƒ^′)(x)=3⋅(sin^2)(5*x+8)⋅cos(5*x+8)⋅(5)

(ƒ^′)(x)=15*(sin^2)(5*x+8)⋅cos(5*x+8)

(B)

Problem 6

u=x2

(u^′)=2*x

v=(cos^2)(x)

(v^′)=2*cos(x)⋅(-sin(x))

(y^′)=2*x⋅(cos^2)(x)-2*cos(x)*sin(x)⋅x2

(y^′)=2*x*(cos^2)(x)-2*x2*cos(x)*sin(x)

Identity: 2*cos(x)*sin(x)=sin(2*x)

(y^′)=2*x*(cos^2)(x)-x2*sin(2*x)

(D)

Problem 7

(y^′)=3*cos(x)-1

(C)

Problem 8

(ƒ^′)(x)=6*x-1/2⋅(1/(-x2))-2*sin(x)

(ƒ^′)(x)=6*x+1/(2*x2)-2*sin(x)

(D)

Problem 9

u=sin(x)+1

(u^′)=cos(x)

v=sin(x)-2

(v^′)=cos(x)

(T^′)(x)=cos(x)*(sin(x)+1)+cos(x)*(sin(x)-2)

Identity: sin(2*x)=2*sin(x)*cos(x)

(T^′)(x)=2*sin(x)*cos(x)+cos(x)-2*cos(x)=sin(2*x)-cos(x)

(C)

Problem 10

(h^′)(θ)=1⋅sin(θ)+(θ+π/2)⋅cos(θ)=sin(θ)+θ*cos(θ)+π/2*cos(θ)

Remember that θ is a variable, like x

(E)

Problem 11

(ƒ^′)(x)=(-sin(x))*sin(x)+cos(x)*(2+cos(x))

(ƒ^′)(π/4)=-1/2+√(,2)/2⋅(2+√(,2)/2)=√(,2)

(C)

Problem 12

(g^′)(x)=(-sin(x)*(sin(x))-cos(x)*(cos(x)+2))/(sin^2)(x)

(g^′)(π/2)=(-1⋅1-0⋅(0+2))/(12)=-1

(B)

Problem 13

(g^′)(x)=2*sin(x)⋅cos(x)-3*sin(3*x)=sin(2*x)-3*sin(3*x)

(C)

Problem 14

(y^′)=2*(sec^2)(2*α-3)

(D)

Problem 15

(ƒ^′)(x)=2*x⋅cot(x)-x2*csc2(x)

(ƒ^′)(π/4)=2⋅π/4⋅1-(π2)/16⋅2=(π*)/8

(A)

Problem 16

(y^′)=-6*sin(2*x)+4*(sec^2)(4*x)

(A)

Problem 17

(g^′)(x)=3*(cos^2)(x)⋅(-sin(x))=-3*(cos^2)(x)*sin(x)

(C)

Problem 18

(h^′)(x)=2*x*sin(x)+x2*cos(x)

(D)

Problem 19

Ralat: (ƒ^′)(π/(2*a))=-1

(ƒ^′)(x)=a*cos(a*x)-b*sin(b*x)

(ƒ^′)(0)=a*cos(0)-b*sin(0)=a=b

∴a=b

(ƒ^′)(π/(2*a))=π/(2*a)*cos(a⋅π/(2*a))-b*sin(b⋅π/(2*b))=-b=-1

∴b=1

∴a+b=1+1=2

(E)

Problem 20

(y^′)=2*x*cos(2*x)-2*x2*sin(2*x)

(A)

Problem 21

u=1-sin(x)

(u^′)=-cos(x)

v=sin(x)-3

(v^′)=cos(x)

(ƒ^′)(x)=(-cos(x)⋅(sin(x)-3)-cos(x)*(1-sin(x)))/((sin(x)-3)2)

(ƒ^′)(x)=(-cos(x)*sin(x)+3*cos(x)-cos(x)+cos(x)*sin(x))/((sin(x)-3)2)

(ƒ^′)(x)=(2*cos(x))/((sin(x)-3)2)

(D)

Problem 22

u=sin(x)

(u^′)=cos(x)

v=sin(x)+cos(x)

(v^′)=cos(x)-sin(x)

(y^′)=(cos(x)*(sin(x)+cos(x))-(cos(x)-sin(x))*sin(x))/((sin(x)+cos(x))2)

(y^′)=(cos(x)*sin(x)+(cos^2)(x)-cos(x)*sin(x)+(sin^2)(x))/((sin(x)+cos(x))2)

(y^′)=((sin^2)(x)+(cos^2)(x))/((sin(x)+cos(x))2)=1/((sin(x)+cos(x))2)=1/(1+sin(2*x))

(A)

Problem 23

(y^′)=2⋅sin(3*x)+2*x⋅3⋅cos(3*x)=2*sin(3*x)+6*x*cos(3*x)

(C)

Problem 24

u=(3*x+4)2

(u^′)=6*(3*x+4)

v=sin(2*x)

(v^′)=2*cos(2*x)

(ƒ^′)(x)=6*(3*x+4)⋅sin(2*x)+(3*x+4)2⋅2*cos(2*x)=*

(ƒ^′)(x)=*

(E)

Problem 25

d(y)/d(x)=8*(sin^3)(2*x+3)⋅cos(2*x+3)

(A)

Problem 26

(ƒ^′)(x)=((cos(x)-sin(x))*(sin(x))-(sin(x)+cos(x))*(cos(x)))/(sin^2)(x)

(ƒ^′)(x)=(cos(x)*sin(x)-(sin^2)(x)-sin(x)*cos(x)-(cos^2)(x))/(sin^2)(x)

(ƒ^′)(π/2)=(-1+0)/1=-1

(B)

Problem 27

ƒ(x)=(3*x2-2)*sin(x2-4)

u=3*x2-2

(u^′)=6*x

v=sin(x2-4)

(v^′)=(2*x)*cos(x2-4)

(ƒ^′)(x)=6*x*sin(x2-4)+(3*x2-2)*(2*x)*cos(x2-4)

(ƒ^′)(x)=6*x⋅(sin(x2-4)+(x2-2/3*x)*cos(x2-4))

(D) (typo?)

Problem 28

(ƒ^′)(x)=5*cos4(4*x-2)*sin(4*x-2)*(-4)

(ƒ^′)(x)=-20*cos4(4*x-2)*sin(4*x-2)

(C)

Problem 29

ƒ(x)=5*(sin(x)*cos(x))=5/2*sin(2*x)

(ƒ^′)(x)=5/2⋅2⋅cos(2*x)=5*cos(2*x)

(B)

Problem 30

(ƒ^′)(x)=x⋅(-2*sin(2*x))+1⋅cos(2*x)

(ƒ^′)(-π/4)=-π/4⋅(-2*sin(-π/2))+1⋅cos(-π/2)=(-π)/2

(A)