In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

23. y = arccsc(secθ), 0<θ<π/2

Evaluate the integrals in Exercises 31–78.

39. ∫(from 0 to π)tan(x/3)dx

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

15. y = sin⁻¹√(1-u²), 0<u<1

Evaluate the integrals in Exercises 31–78.

31. ∫e^x sin(e^x)dx

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

88. lim(x→0) (tan x)/(x + sin(x))

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

17. y = ln(arccos(x))

Evaluate the integrals in Exercises 31–78.

49. ∫x3^(x²)dx

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

97. lim(x→0) (10^x - 1)/x

Evaluate the integrals in Exercises 31–78.

37. ∫(from -1 to 1)dx/(3x-4)

In Exercises 79–84, solve for y.

83. ln(y-1) = x + ln(y)

113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

1. y = 10e^(-x/5)

133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

93. lim(x→0) (csc(x) - cot(x))

Use l’Hôpital’s Rule to find the limits in Exercises 85–108.

102. lim(x→0) (x sin(x²))/(tan³x)

Evaluate the integrals in Exercises 31–78.

58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ

Evaluate the integrals in Exercises 31–78.

64. ∫(from 1 to e)(8ln3 log_3(θ))/θ dθ

In Exercises 79–84, solve for y.

81. 9e^(2y) = = x^2

In Exercises 129–132 solve the initial value problem.

129. dy/dx = e^(-x-y-2), y(0) = -2

Evaluate the integrals in Exercises 31–78.

67. ∫(from -2 to 2)3dt/(4+3t²)

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

c. f(x) = 10x^3 + 2x^2, g(x) = e^x

Evaluate the integrals in Exercises 31–78.

47. ∫(1/r)csc²(1+ln(r))dr

23. What roles do the functions e^x and ln(x) play in growth comparisons?

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

a. f(x) = x, g(x) = x², (a, b) = (-2, 0)

4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

a. x² + √x

21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).

In Exercises 41–44:

a. Find f⁻¹(x).

41. f(x) = 2x + 3, a = −1

a. Show that h(x) = x³ / 4 and k(x) = (4x)^(1/3) are inverses of one another.

In Exercises 41–44:

a. Find f⁻¹(x).

43. f(x) = 5 − 4x, a = 1/2

82. Use the definitions of the hyperbolic functions to find each of the following limits.

a. lim(x→∞) tanh x