hyperbolic table midterm 1

Question 1

what is sinh(x)
domain and range
derivative
inverse

Answer 1

(e^x - e^-x)/2
d: (-∞, +∞)
r: (-∞, +∞)
d/dx= cosh(x)
arcsinh(x)

Question 2

what is cosh(x)
domain and range
derivative
inverse

Answer 2

(e^x + e^-x)/2
d: (-∞, +∞)
r: [1, ∞)
d/dx= sinh(x)
arccosh(x)

Question 3

what is tanh(x)
domain and range
derivative
inverse

Answer 3

(e^x - e^-x)/(e^x +e^-x)
d: (-∞, +∞)
r: (-1, 1)
d/dx= sech²(x)
arctanh(x)

Question 4

what is coth(x)
domain and range
derivative
inverse

Answer 4

(e^x +e^-x)/(e^x - e^-x)
d: (-∞, 0) U (0, + ∞)
r: (-∞, -1) U (1, + ∞)
d/dx= -csch²(x)
arccoth(x)

Question 5

what is sech(x)
domain and range
derivative
inverse

Answer 5

2/(e^x +e^-x)
d: (-∞, + ∞)
r: (0,1]
d/dx= -sech(x)tanh(x)
arcsech(x)

Question 6

what is csch(x)
domain and range
derivative
inverse

Answer 6

2/(e^x -e^-x)
d: (-∞, 0) U (0, + ∞)
r: (-∞, 0) U (0, + ∞)
d/dx= -csch(x)coth(x)
arccsch(x)

Question 7

derivative of arcsinh(x)

Answer 7

1/√ (x²+1)

Question 8

derivative of arccosh(x)

Answer 8

1/√ (x²-1)

Question 9

derivative of arctanh(x)

Answer 9

1/(1-x²)

Question 10

derivative of arccoth(x)

Answer 10

1/(1-x²)

Question 11

derivative of arcsech(x)

Answer 11

-1/(x√(1-x²))

Question 12

derivative of arccsch(x)

Answer 12

-1/(|x|√(x²+1))