differentialilaskut

Differentialilaskut, or differential calculus, is a branch of mathematics that focuses on rates of change and the slopes of curves. It provides tools for describing how small changes in one quantity affect another, using derivatives and differentials as central concepts.

The derivative at a point measures the instantaneous rate of change as the input varies. It is defined as a limit of average rates of change and serves as the basis for linear approximation: the function is locally approximated by a tangent line. Differentials, denoted dx and dy, encode infinitesimal changes and enable differential relationships between variables. Core rules include the chain rule for composition, the product and quotient rules, and higher-order derivatives. Taylor's theorem shows how a function can be approximated by polynomials around a point.

Historically, differential calculus emerged in the 17th century from the work of Gottfried Wilhelm Leibniz and Isaac Newton. Notation such as dy/dx arose from their development. Over the 18th and 19th centuries, the subject was made rigorous through the formal notion of limits and the development of analysis by Cauchy, Weierstrass, and others.

Applications span physics, engineering, economics, biology, and data science. Derivatives describe velocity and acceleration, optimize profits, model growth rates, and form the basis of differential equations that describe dynamic systems. Numerical methods approximate derivatives and solve differential equations when closed-form solutions are unavailable.

Differential calculus is complemented by integral calculus, with the fundamental theorem of calculus linking derivatives and integrals. Together, they constitute the broader field of infinitesimal calculus, enabling both local linear approximations and accumulation across intervals.