For sufficiently smooth functions on the real line, the Legendre transform f ∗ ∗ of a function f can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as D f ( ⋅ ⋅ ) = ( D f ∗ ∗ ) − − 1 ( ⋅ ⋅ ) , where D is an operator of differentiation, ⋅ ⋅ represents an argument or input to the associated function, ( ϕ ϕ ) − − 1 ( ⋅ ⋅ ) is an inverse function such that ( ϕ ϕ ) − − 1 ( ϕ ϕ ( x ) ) = x ,