In Exercises 5–8, show that each function is a solution of the given initial value problem.
7. Differential Equation: xy' + y = -sin(x), x>0
Initial condition: y(π/2) = 0
Solution candidate: y = cos(x)/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 + 3) / (x − 2)

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

37. y=s√(1-s²) + arccos(s)

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) = (1 − x)³

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

30. lim (θ → 0) ((1/2)^θ - 1) / θ

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.

75. lim (x → ∞) e^(x²) / (x e^x)

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

13. y = 6sinh(x/3)