On a property of the Fenchel transform

We consider the class of functions $\Phi \colon \mathbb{R} \to [0, + \infty]$, which are lower semicontinuous, even, convex and $ \Phi (0) = 0 $. The Fenchel transform $\Psi$ from $\Phi$ also belongs to this class of functions. We will define functions that play the role of derivatives for all functions from our class and prove that these functions are mutually inverse in a generalized sense.