A type of Inconsistent Mathematics structured on Paraconsistent Logic (PL) and that has, as the main purpose, the study of common mathematical objects such as sets, numbers and functions, where some contradictions are allowed, is called Paraconsistent Mathematics. The PL is a non-Classical logic and its main property is to present tolerance for contradiction in its fundamentals without the invalidation of the conclusions. In this paper (part 1), we use the PL in its annotated form, denominated Paraconsistent Annotated Logic with annotation of two values—PAL2v for present a first-order Paraconsistent Derivative. The PAL2v has, in its representation, an associated lattice FOUR based on Hasse Diagram. This PAL2v-Lattice allows development of a Para-consistent Differential Calculus based on fundamentals and equations obtained by geometric interpretations. In this first article it is presented some examples applying derivatives of first-order with the concepts of Paraconsistent Mathematics. In the second part of this work we will show the Paraconsistent Derivative of second-order with application examples.
 D’Ottaviano, I.M.G. and Carvalho, T.F. (2005) Da Costa’s Paraconsistent Differential Calculus and the Transference Theorem. 2nd Indian International Conference on Artificial Intelligence (II CAI-05), Pune, 1659-1678.
 Jas’kowski, S. (1969) Propositional Calculus for Contradictory Deductive Systems. Studia Logica, 24, 143-157.
 Da Silva Filho, J.I. (2011) Paraconsistent Annotated Logic in Analysis of Physical Systems: Introducing the Paraquantum hψ Factor of Quantization. Journal of Modern Physics, 2, 1397-1409. http://dx.doi.org/10.4236/jmp.2011.211172
 Da Costa, N.C.A. (2000) Paraconsistent Mathematics. In: Batens, D., Mortensen, C., Priest, G., van Bendegen, J.P., Eds., Frontiers in Paraconsistent Logic: Proceedings. King’s College Publications, London, 165-179.
 Da Silva Filho, J.I. (2011) Paraconsistent Annotated Logic in Analysis of Physical Systems: Introducing the Paraquantum γψ Gamma Factor. Journal of Modern Physics, 2, 1455-1469. http://dx.doi.org/10.4236/jmp.2011.212180
 Arruda, A.I. (1989) Aspects of the Historical Development of Paraconsistent Logic. In: Priest, G., Routley, R. and Norman, J., Eds., Paraconsistent Logic: Essays on the Inconsistent, Philosophia Verlag, 99-130.
 Da Costa, N.C.A. (1974) On the Theory of Inconsistent Formal Systems. Notre Dame Journal of Formal Logic, 15, 497-510.
 Da Silva Filho, J.I. (2012) Analysis of the Emissions Spectral line of the Paraquantum with Hydrogen Atom. Journal of Modern Physics, 3, 233-254.
 Da Silva Filho, J.I. (2012) An Introductory Study of the Hydrogen Atom with Paraquantum Logic. Journal of Modern Physics, 3, 312-333.
 Da Silva Filho, J.I. (2012) Study of the Interactions between Particles Based in Paraquantum Logic. Journal of Modern Physics, 3, 362-376.
 Diethelm, K. and Ford, N. (2004) Multi-Order Fractional Differential Equations and Their Numerical Solution. Applied Mathematics and Computation, 154, 621-640.