dy

In mathematics, the symbol dy denotes the differential of the dependent variable y, typically when y is a differentiable function of an independent variable x. It represents an infinitesimal change in y associated with a small change in x.

If y = f(x), then dy = f'(x) dx. For small increments dx, the actual change Δy is approximately dy, a relation used in linear approximation and the chain rule.

In integrals, dy is the differential element used when integrating with respect to y, as in ∫ g(y) dy. In physics and engineering, dy appears in rate expressions such as dy/dt and serves as a differential form basis alongside dx in more general contexts, including line and surface integrals.

Within higher-dimensional spaces, dy can be treated as a one-form that, in coordinates, acts on tangent vectors to yield directional changes in y. It is one of the basic differentials that together with dx, dz, etc., form the exterior calculus used in multivariable calculus and differential geometry.

Leibniz introduced the differential notation in the 17th century, and dy remains standard in mathematical analysis and applied disciplines.

See also: derivative, differential, differential form, chain rule, integral.