dx

dX denotes the differential of a variable X or the infinitesimal change in X. The exact meaning depends on the mathematical setting. In single-variable calculus, if X is a differentiable function of a parameter t, then dX = (dX/dt) dt, representing the small change in X corresponding to an infinitesimal change in t. For a vector-valued X(t) = (X1(t), ..., Xn(t)), the differential is dX = (dX1, ..., dXn), with each dXi = (dXi/dt) dt in the simple one-parameter case.

In stochastic calculus, dX_t denotes the infinitesimal increment of a stochastic process X_t and is treated as a formal differential obeying Ito rules. For example, if X_t = μ t + σ W_t, then dX_t = μ dt + σ dW_t, where W_t is standard Brownian motion, and (dW_t)^2 = dt. In such contexts, cross terms like dt dW_t are zero in Ito calculus.

In differential geometry, the differential of a smooth map X: M -> N at a point p, denoted dX_p, is a linear map between the tangent spaces T_pM and T_{X(p)}N. When X represents a coordinate function, its differential components dX_i describe infinitesimal changes along coordinate directions.

Because dX is context dependent, it is best interpreted with the surrounding notation. It is a standard tool across calculus, probability, physics, and geometry to model infinitesimal changes, increments, or linearizations of X.