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The purpose of this work is to show that the Khalil and Katagampoulaconformable derivatives are equivalent to the simple change of variables $x$$\rightarrow $ $x^{\alpha }/\alpha ,$ where $\alpha $ is the order of thederivative operator, when applied to differential functions. Although thismeans no \textquotedblleft new mathematics\textquotedblright\ is obtained byworking with these derivatives, it is a second purpose of this work to arguethat there is still significant value in exploring the mathematics and physicalapplications of these derivatives. This work considers linear differentialequations, self-adjointness, Sturm-Liouville systems, and integral transforms.A third purpose of this work is to contribute to the physical interpretationwhen these derivatives are applied to physics and engineering. Quantummechanics serves as the primary backdrop for this development.

On the nature of the conformable derivative and its applications to physics