For Exercises 127 and 128 find a function f satisfying each equation.
127. ∫₂ˣ √(f(t)) dt = x ln x

Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.

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

y = e^(5-7x)

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

29. y = (1 - t)coth⁻¹(√t)

Indeterminate Powers and Products

Find the limits in Exercises 53–68.

55. lim (x → ∞) (ln x)^(1/x)

Solve the initial value problems in Exercises 87 and 88.

88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1

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

27. lim (x → (π/2)^-) (x - π/2) sec x