Differentialquotient

Differentialquotient is a term used in calculus to denote the quotient that forms the basis for the derivative. For a function f defined on an interval around a point x, the differential quotient at x is the ratio [f(x + h) − f(x)]/h for h ≠ 0. If the limit of this quotient exists as h approaches 0, the limit is the derivative f′(x). Thus the differential quotient represents the instantaneous rate of change of f with respect to x and, when it exists, equals the slope of the tangent line to the graph of f at that point.

An example helps illustrate the concept. Let f(x) = x^2. The differential quotient at a point x is [(x + h)^2 − x^2]/h = (2xh + h^2)/h = 2x + h, which tends to 2x as h → 0. Therefore f′(x) = 2x, and the differential quotient approaches the slope of the tangent line at x.

The differential quotient has a close relationship with the differential. If f is differentiable at x, the differential dy = f′(x) dx, and the differential quotient dy/dx equals f′(x). Geometrically, the differential quotient measures the slope of the curve at a point and is undefined at points where the derivative does not exist, such as f(x) = |x| at x = 0, where the left and right limits of the quotient differ.

Generalizations include extensions to multiple variables, where partial differential quotients and directional derivatives describe rates of change with respect to a chosen variable or direction. In calculus, the differential quotient is a foundational concept linking average and instantaneous rates of change.