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

83. y = 3^(log₂ t)

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))

Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.

f(x) = 27x³

22. The function ln x grows slower than any polynomial Show that ln(x) grows slower as x→∞ than any nonconstant polynomial.

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

44. lim (x → 0⁺) (csc x - cot x + cos x)

Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.

f(x) = (x + b) / (x − 2), b > −2 and constant

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

19. y = (sech θ)(1-ln(sech θ))

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

Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.

2. sinh x = 4/3

Evaluate the integrals in Exercises 39–56.

41. ∫2y dy/(y²-25)

In Exercises 13–24, find the derivative of y with respect to the appropriate variable.

21. y = ln(cosh v) - 1/2 tanh²v

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

14. lim (t → 0) sin 5t / 2t

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

y = ln(e^(θ)/(1+e^θ))

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

105. lim(x→∞) x arctan(2/x)

Verify the integration formulas in Exercises 37–40.

39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

In Exercises 27–32, find dy/dx.

e^(2x)=sin(x+3y)

In Exercises 27–32, find dy/dx.

3+siny = y-x^3

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

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

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

59. y = 2^x

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

25. y = ln(ln(x))

Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.

6. sinh(2ln x)

Evaluate the integrals in Exercises 91–102.

93. ∫(arcsin x)²dx/√(1-x²)

Solve the differential equation in Exercises 9–22.

19. y²(dy/dx) = 3x²y³ - 6x²

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

73. y = log₄ x + log₄ x²

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

39. lim (x → ∞) (ln 2x - ln(x + 1))

15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.

13. For what x>0 does x^(x^x) = (x^x)^x? Give reasons for your answer.

19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.

Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?