Let ℝ denote the set of all real numbers. Let f : ℝ → ℝ and g : ℝ → (0, 4) be functions defined by f(x) = logₑ(x² + 2x + 4) and g(x) = 4/(1 + e⁻²ˣ). Define the composite function f ∘ g⁻¹ by (f ∘ g⁻¹)(x) = f(g⁻¹(x)), where g⁻¹ is the inverse of the function g. Then the value of the derivative of the composite function f ∘ g⁻¹ at x = 2 is _______________.