Modul 2: Derivator

Question 1

d/dx x^n =

Answer 1

n x^n-1

Question 2

d/dx e^kx =

Answer 2

ke^kx

Question 3

d/dx lnx =

Answer 3

1/x

Question 4

d/dx sinx =

Answer 4

cosx

Question 5

d/dx sin(kx)=

Answer 5

kcos(kx)

Question 6

d/dx cosx =

Answer 6

-sinx

Question 7

d/dx cos(kx)=

Answer 7

-ksin(kx)

Question 8

d/dx rotx =

Answer 8

1/2 rotx

Question 9

d/dx 1/x=

Answer 9

-1/x^2

Question 10

Skriv om rotx

Answer 10

X^1/2

Question 11

Skriv om xrotx

Answer 11

X^3/2

Question 12

Skriv om 1/x

Answer 12

x^-1

Question 13

Skriv om 1/x^n

Answer 13

x^-n

Question 14

Derivera y=rotx

Answer 14

1/2rotx

Question 15

Kedjeregeln
Derivera y=sin(x^2)

Answer 15

y’ = cos (x^2) * 2x

Question 16

Derivera
ln(cos2x)

Answer 16

y’= (1/cos2x) * -2sin2x
= -2sin2x/cos2x
=-2tan2x

Question 17

Vad är produktregeln

Answer 17

Låt f =f(x) och g = g(x)
fg)’ = f’g + f*g’

f(x) = u(x) * v(x)
f’(x) = u’(x) * v(x) + u(x) * v’(x)

Question 18

h(x) = 3x^2 * sin(-4x)

Answer 18

h’(x) = 6xsin(-4x) - 12x^2 *cos(-4x)

Hint
(-12= -4 *3)

Question 19

Vad är kvotregeln

Answer 19

(f/g)’ = f’g - fg’ /g^2

f(x) = g(x)/h(x)
f’(x) = [h(x) * g’(x) - g(x)*h’(x)] / [h(x)]^2

Question 20

Derivera h(x) = tan x = sinx/cosx

Answer 20

h’(x) = cosx *cosx - sinx (-sinx) / (cos^2 (x))
= cos^2 (x) +sin^2 (x) / cos^2 (x)

Två sätt att skriva svaret på:
Trig ettan: h’(x) = 1/cos^2 (x)
Bråkuppdelning: cos^2(x)/cos^2(x) + sin^2(x)/sin^2(x) = 1 + tan^2(x)

Question 21

d/dx arcsinx =

Answer 21

1/rot(1-x^2)

Question 22

d/dx arccosx =

Answer 22

-1/rot(1-x^2)

Question 23

d/dx arctanx =
Hur ser grafen ut?

Answer 23

1/(1+x^2)

lim_x->○○ arctanx = π/2
lim_x->-○○ arctanx = -π/2

Question 24

Skriv om sin(-2x)

Answer 24

2sinxcosx

Question 25

Skriv om sinxcosx

Answer 25

1/2 sin(2x)

Question 26

Skriv om Sin(-a)

Answer 26

-sin(a)

Question 27

Skriv om cos(-a)

Answer 27

cos(a)

Question 28

d/dx e^x

Answer 28

e^x

Question 29

d/dx tanx

Answer 29

1+tan^2_x

Question 30

d/dx arctanx

Answer 30

1/(1+x^2)

Question 31

När är cosx = sinx

Answer 31

X= π/4 + 2πη
X= 5π/4 + 2πη
Där n är heltal Zz