The term "differential coefficient" is a historical name for what is now called the derivative of a function. If f is a real-valued function of a real variable, the differential df and the differential dx are related by df = f'(x) dx, where f'(x) is the derivative or the differential coefficient at x. In this language the differential coefficient is the multiplicative factor that converts the infinitesimal change dx into the corresponding linearized change df. For background on the wider subject, see calculus.
Definition and formal meaning
Formally, the derivative at a point x is defined as the limit of the difference quotient, when that limit exists: f'(x) = lim_{h->0} (f(x+h)-f(x))/h. In older writings, the value f'(x) would be called the differential coefficient of the function f(x). Equivalently, when f is differentiable at x the differential df can be written as df = (differential coefficient) · dx, emphasizing the linear relationship between infinitesimal increments.
Properties, rules and examples
The differential coefficient inherits the familiar rules of differentiation: it is linear, satisfies the product and quotient rules, and follows the chain rule for compositions of functions. If the differential coefficient is constant over an interval, the original function is affine (linear up to a constant): constant differential coefficient ⇔ function of the form ax + b. Conversely, for a linear function the differential coefficient equals its slope. Basic examples include polynomials, exponentials and trigonometric functions, for which the differential coefficient can be computed by standard formulas.
- Interpretation: the differential coefficient measures instantaneous rate of change (for example, velocity as the rate of change of position).
- Continuity: existence of a differential coefficient at a point implies the function is continuous there.
- Higher derivatives: repeated application yields second and higher differential coefficients (second derivative, etc.).
Historical use and terminology
"Differential coefficient" appears in older calculus texts and was common when differentials and infinitesimals were the preferred language. Modern texts usually prefer "derivative" and symbolic notations such as f'(x) or df/dx (Leibniz notation). Notable historical treatments that used older wording include early editions of Silvanus P. Thompson's popular book Calculus Made Easy, where later editors and commentators replaced the phrase with "derivative" to match contemporary usage.
Modern interpretation and extensions
Contemporary mathematics often treats the differential as a linear map: the differential at x is the best linear approximation to f near x, and its matrix representation (in one variable) is the differential coefficient. This viewpoint extends naturally to functions of several variables, where partial derivatives and the derivative as a linear transformation (Jacobian) replace the single differential coefficient. For further reading on these concepts see discussions of the coefficient, constant cases, and the general notion of linear function and differentiability in standard calculus texts or reference notes linked from differential and calculus.
Although the phrase "differential coefficient" has fallen out of favor, it remains useful to recognize the historical terminology when reading older literature or when connecting the intuitive infinitesimal language (df and dx) with the limit-based definition of the derivative. For commentary on terminology changes and modern edits, see the discussion by editors such as Martin Gardner referenced in historical updates to classic works (Gardner on Thompson).
For concise summaries of related ideas and rules, introductory chapters of modern calculus texts or authoritative reference notes provide systematic accounts of differentiation techniques, proofs of key properties, and examples illustrating the role of the differential coefficient in applications across physics, engineering and economics. See also introductory guides to functions and calculus for step-by-step development.