Calculus Without Limits

This paper presents an algebraic and geometric-functional approach
to introducing the derivative for elementary functions without using
limits. The derivative is defined as a functional correspondence
between the abscissa of a point on the graph of a function and
the slope of the unique tangent line drawn at that point (the X-K
correspondence). The method is developed systematically starting
from single-variable polynomial functions by introducing the notions
of multiple roots and tangency through an algebraic condition of
repeated intersection. On this foundation, the derivative function
is constructed and key differentiation rules are established,
including the sum, product, quotient, and composite function
rules. The approach is then extended to rational power functions,
exponential functions, logarithmic functions with an arbitrary base,
and trigonometric functions, yielding the same derivative formulas
as in classical analysis. Finally, the increment of a function and the
differential are interpreted geometrically via the tangent line, and
the classical limit definition of the derivative arises as an analytical
formalization of this geometric differential. The results demonstrate
both mathematical consistency and strong pedagogical potential
for secondary and undergraduate instruction.