What does dx mean in derivative notation?

In the derivative notation dy/dx, the symbol 'dx' represents an infinitesimal change in the independent variable x. It is a foundational component of Leibniz's notation, which conceptualizes the derivative as the ratio of two infinitesimally small quantities: the differential dy (the corresponding change in the function's output) and the differential dx. This notation is profoundly intuitive because it directly suggests the derivative's core idea as an instantaneous rate of change, computed as the limit of the ratio Δy/Δx as Δx approaches zero. While 'dx' originates from this historical notion of an infinitesimal, its rigorous modern interpretation is defined through the limit process; it is not a number itself but a part of a symbolic whole that denotes the operation of differentiation with respect to x.

The true power and enduring legacy of the 'dx' notation lie in its operational utility and suggestive properties, which far exceed those of prime notation (f'(x)). It explicitly states the variable of differentiation, which is crucial for functions involving multiple variables and for applying the chain rule. The chain rule, expressed as dy/dt = (dy/dx)*(dx/dt), becomes a natural algebraic manipulation of differentials, a mnemonic that correctly reflects the underlying theorem. Furthermore, the notation seamlessly integrates with the language of differentials in integral calculus, where 'dx' serves as the differential in the integrand, specifying the variable of integration and conceptually representing an infinitesimal width in the context of Riemann sums. This creates a unified notational framework across differential and integral calculus.

From a more advanced mathematical perspective, 'dx' can be granted an independent meaning as a differential form, particularly in the context of multivariable calculus and differential geometry. In this sophisticated framework, dx is a linear functional (a one-form) that projects a vector onto its component in the x-direction. This allows for a rigorous, coordinate-independent treatment of calculus on manifolds. However, in single-variable calculus, it is most prudent to treat dy/dx as a single, indivisible symbol for the derivative, while acknowledging that the separate parts 'dy' and 'dx' can be meaningfully defined in a controlled way through the concept of the differential. The differential 'dx' is defined as an arbitrary real number increment, and the differential 'dy' is then defined as f'(x) dx, providing a linear approximation to the actual change Δy.

The practical implication of this notation is that it is indispensable for applied sciences and engineering, where the manipulation of derivatives as ratios—such as in separation of variables for differential equations or related rates problems—is a standard computational technique. The notation's design encourages correct intuitive reasoning about rates and changes, even though the foundational justification rests on limits. Therefore, 'dx' is far more than historical relic; it is a semantically rich symbol that denotes the variable of change, enables powerful heuristic manipulations, and provides a critical bridge to integral calculus and higher-dimensional analysis, all while being rooted in a precise limit-based definition.