In Exercises 27–32, find dy/dx.
ln y = e^y sinx

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

47. y=(arccot(x³))³

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

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)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

60. lim (x → 0) (e^x + x)^(1/x)

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 139–142, find the length of each curve.

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