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.
119. y = (sin x)ˣ

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

10. y = ln(t^(3/2))

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

y = xe^x-e^x

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

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

15. lim(x→∞)arctan(x)

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x

L’Hôpital’s Rule

Find the limits in Exercises 103–110.

108. lim(x→∞)(e^x arctan(e^x))/(e^(2x)+x)