In Exercises 5 and 6, solve for t.

6. ln(t-2) = ln8 - ln(t)

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 111–114.

112. ∫₁^(eˣ) (1 / t) dt

Find the values in Exercises 9–12.

11. tan(arcsin(-1/2))

For Exercises 127 and 128 find a function f satisfying each equation.

128. f(x) = e² + ∫₁ˣ f(t) dt

Solve the differential equation in Exercises 9–22.

12. (dy/dx) = 3x²e^(-y)

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

25. y=arcsec(2s+1)

11. Show that if positive functions f(x) and g(x) grow at the same rate as x→∞, then f=O(g) and g=O(f).

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.

117. y = (√t)ᵗ

47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?

130. Use the identity arccot(u)=π/2 - arctan(u) to derive the formula for the derivative of arccot(u) in Table 7.4 from the formula for the derivative of arctan(u).

Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.

f(x)=x²+1, x≥0

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

y = e^(θ)(sinθ + cosθ)

Evaluate the integrals in Exercises 97–110.

101. ∫ (log₁₀x / x) dx

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

10. y = ln(t^(3/2))

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

81. y = log₁₀(e^x)

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

Evaluate the integrals in Exercises 53–76.

67. ∫dx/(2+(x-1)²)

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.

63. tanh⁻¹(-1/2)

Evaluate the integrals in Exercises 111–114.

113. ∫₁^(1/x) (1 / t) dt, x > 0

Evaluate the integrals in Exercises 77–90.

79. ∫(from -1 to 0)6dt/√(3-2t-t²)

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

19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)

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

Solve the initial value problems in Exercises 87 and 88.

88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1

For Exercises 127 and 128 find a function f satisfying each equation.

127. ∫₂ˣ √(f(t)) dt = x ln x

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

55. lim (x → ∞) (ln x)^(1/x)

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)

Evaluate the integrals in Exercises 31–78.

73. ∫dx/√(-2x-x²)

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))