Refer to the function gd(y) = tan⁻¹(sinh y), called the Gudermannian. In a map of the earth constructed by Mercator projection, points located y radial units from the equator correspond to points on the globe of latitude gd(y).

Let ƒ(y) = 2 tan⁻¹(ey) − π/2. Prove that gd(y) = ƒ(y). Show that gd'(y) = ƒ'(y) and ƒ'(0) = g(0).