3 ideas
A limit describes the value a function approaches as its input approaches a point, defined rigorously by the epsilon-delta condition. Both the derivative and the definite integral are constructed as limits, making the limit the single primitive from which the whole subject is built.
The derivative is the limit of the difference quotient as the interval shrinks to zero, geometrically the slope of the tangent line. Differentiation rules (product, quotient, chain) let this be computed mechanically for composite functions rather than from the limit each time.
The Fundamental Theorem of Calculus shows differentiation and integration are inverse operations: the definite integral of a rate recovers the accumulated total, and the derivative of an accumulation function returns the integrand. This is what makes integrals computable via antiderivatives rather than Riemann sums.