15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

66. lim (x → 0⁺) x (ln x)²

7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.

a. lim(x→∞)A(t)

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

61. y = √(θ + 3) sin θ

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

64. y = 1/(t(t+1)(t+2))

20. Solid of revolution The region between the curve y=1/(2√x) and the x-axis from x=1/4 to x=4 is revolved about the x-axis to generate a solid.

b. Find the centroid of the region.

Find the limits in Exercises 1–6.

1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)