For a differentiable convex function f on the real line with the first derivative f ′ and its inverse ( f ′ ) − − 1 , the Legendre transform of f , f ∗ ∗ , can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other, i.e., f ′ = ( ( f ∗ ∗ ) ′ ) − − 1 and ( f ∗ ∗ ) ′ = ( f ′ ) − − 1 .