Use l’Hôpital’s rule to find the limits in Exercises 7–52.
41. lim (x → 0⁺) (ln x)² / ln(sin x)

Evaluate the integrals in Exercises 33–54.

∫₀^(π/4) (1 + e^(tan θ)) sec²θ dθ

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

9. (sinh(x)+cosh(x))⁴

130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?

25. First-order chemical reactions In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-gluconolactone into gluconic acid, for example,

dy/dt = -0.6y

when t is measured in hours. If there are 100 grams of δ-gluconolactone present when t=0, how many grams will be left after the first hour?

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

65. y = (cos θ)^(√2)

Solve the differential equation in Exercises 9–22.

15. √x (dy/dx) = e^(y+√x), x > 0