a. By writing y = x^x in the form ln y = x ln x, show that dy/dx = x^x (ln x + 1).

b. Show that the function f(x) = x^x − 2 has a root, α, in the interval [1.4, 1.6].

c. Taking x₀ = 1.5 as a first approximation to α, apply the Newton–Raphson procedure once to obtain a second approximation to α, giving your answer to 4 decimal places.

d. By considering a change of sign of f(x) over a suitable interval, show that α = 1.5596, correct to 4 decimal places.