We present a theoretical framework—DIM (Infinitesimal Dynamics of the Spacetime Matrix)—in which time, gravity, gauge fields, and matter emerge from derivatives of a single scalar field Φ(χ, xμ), defined over a log-periodic fractal extension of spacetime. In this model, time is the first derivative with respect to χ, gravity the second, and higher-order modulations produce gauge and fermionic phenomena. The system evolves from a maximally symmetric state, the Derivative Vacuum Φ0, where all derivatives vanish. Using a 5D Klein-Gordon equation and renormalization group analysis, we show asymptotic safety and infrared stabilization. Numerical simulations confirm multifractal structure and vibrational condensates. The model offers falsifiable predictions: deviations in Newtonian gravity at small scales, anomalous gravitational wave modes, and fractal anisotropies in the cosmic microwave background. DIM provides a mathematically consistent and ontologically grounded route to unifying physics from a single scalar source.