derivatives & domain and range

Question 1

dy/dx (sinx)

Answer 1

cos(x)

Question 2

dy/dx (cosx)

Answer 2

-sin(x)

Question 3

dy/dx (tanx)

Answer 3

(sec^2)(x)

Question 4

dy/dx (cscx)

Answer 4

-cot(x)csc(x)

Question 5

dy/dx (secx)

Answer 5

tan(x)sec(x)

Question 6

dy/dx (cotx)

Answer 6

-(csc^2)(x)

Question 7

dy/dx (e^x)

Answer 7

e^x

Question 8

dy/dx [a^x]

Answer 8

ln(a) * a^x

Question 9

dy/dx [ln(x)]

Answer 9

1/x

Question 10

dy/dx [log(base b)(x)]

Answer 10

[1/(ln(b))] * (1/x)

Question 11

dy/dx (arcsin(x))

Answer 11

1/√(1-x^2)

Question 12

dy/dx (arccos(x))

Answer 12

-1/√(1-x^2)

Question 13

dy/dx (arctan(x))

Answer 13

1/(1+x^2)

Question 14

dy/dx (arccsc(x))

Answer 14

-1/(abs(x)√(x^2 - 1)) , abs(x) > 1

Question 15

dy/dx (arcsec(x))

Answer 15

1/(abs(x)√(x^2 - 1)) , abs(x) > 1

Question 16

dy/dx (arccot(x))

Answer 16

-1/(1+x^2)

Question 17

domain and range of cos(x)

Answer 17

domain: all real numbers
range: [-1,1]

Question 18

domain and range of sin(x)

Answer 18

domain: all real numbers
range: [-1,1]

Question 19

domain and range of tan(x)

Answer 19

domain: x ≠ (2n-1)(π/2)
range: all real numbers

Question 20

domain and range of cot(x)

Answer 20

domain: x ≠ nπ
range: all real numbers

Question 21

domain and range of csc(x)

Answer 21

domain: x ≠ nπ
range: (-∞ , -1] U [1, ∞)

Question 22

domain and range of sec(x)

Answer 22

domain: x ≠ (2n-1)(π/2)
range: (-∞ , -1] U [1, ∞)

Question 23

domain and range of arccos(x)

Answer 23

domain: [-1,1]
range: [0,π]

Question 24

domain and range of arcsin(x)

Answer 24

domain: [-1,1]
range: [-π/2 , π/2]

Question 25

domain and range of arctan(x)

Answer 25

domain: all real numbers
range: (-π/2 , π/2)

Question 26

domain and range of arccsc(x)

Answer 26

domain: (-∞ , -1] U [1, ∞)
range: [-π/2 , π/2], y ≠ 0

Question 27

domain and range of arccot(x)

Answer 27

domain: all real numbers
range: (0,π)

Question 28

domain and range of arcsec(x)

Answer 28

domain: (-∞ , -1] U [1, ∞)
range: [-π/2 , π/2], y ≠ π/2