Solve the initial value problems in Exercises 55–58.

57. d²y/dx² = 2e^(−x), y(0) = 1, y′(0) = 0

"In Exercises 59–86, find the derivative of y with respect to the given independent variable.

67. y = 7^(sec θ) ln 7"

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(5-7x)

In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.

120. y = x^(sin x)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

29. y = (1 - t)coth⁻¹(√t)

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.

49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

27. y = (1 - θ)tanh⁻¹(θ)

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

18. y = t√(ln t)

Evaluate the integrals in Exercises 53–76.

63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

105. lim(x→∞) x arctan(2/x)

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = cos(e^(-θ^2))

Evaluate the integrals in Exercises 53–76.

59. ∫(from 0 to 1)4ds/√(4-s²)

Evaluate the integrals in Exercises 97–110.

105. ∫₀² (log₂(x + 2) / (x + 2)) dx

In Exercises 139–142, find the length of each curve.

139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.

1. lim (x → -2) (x + 2) / (x² - 4)

Evaluate the integrals in Exercises 39–56.

45. ∫(from 1 to 2)(2ln x)/x dx

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

13. lim(x → 1⁻)arcsin(x)

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

85. y = log₂(8t^(ln 2))

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.

51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

35. y=arccsc(e^t)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

33. y = csch⁻¹(1/2)^θ

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

32. lim (x → 0) (3^x - 1) / (2^x - 1)

Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).

In Exercises 139–142, find the length of each curve.

141. y = ln(cos(x)) from x = 0 to x = π/4.

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

65. sech⁻¹(3/5)

13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))

Evaluate the integrals in Exercises 39–56.

49. ∫3sec²t/(6 + 3tan(t)) dt

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

83. y = 3^(log₂ t)

Use l’Hôpital’s rule to find the limits in Exercises 7–52.

39. lim (x → ∞) (ln 2x - ln(x + 1))