In Exercises 5–8, show that each function is a solution of the given initial value problem.
5. Differential Equation: 2y + y' = 4x + 2
Initial condition: y(-1) = e² - 2
Solution candidate: y = e^(-2x) + 2x

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

47. y=(arccot(x³))³

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)

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

29. y = ln(1/(x√(x+1)))

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

60. lim (x → 0) (e^x + x)^(1/x)

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 27–32, find dy/dx.

ln y = e^y sinx