Differentiointi

Differentiointi is the process of determining the derivative of a function, describing the instantaneous rate of change with respect to its input. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point. The concept is central to differential calculus and underpins many mathematical analyses and applications.

Formally, if f is defined on an interval around x, the derivative f'(x) is defined as the limit lim h→0 [f(x+h) - f(x)] / h, provided the limit exists. Derivatives are denoted in several notations, such as f'(x), dy/dx, or Df. Common differentiation rules include the power rule, product rule, quotient rule, and chain rule. The derivative of a constant is zero.

Examples include d/dx x^2 = 2x, d/dx sin x = cos x, d/dx e^x = e^x, and d/dx ln x = 1/x. The second derivative f''(x) provides information about concavity and the acceleration of a function; higher-order derivatives describe further rates of change.

Applications span physics (velocity is the derivative of position; acceleration is the second derivative), engineering, economics (marginal analysis), biology, and statistics. In optimization, derivatives are used to locate maxima and minima by solving f'(x) = 0 and examining second derivatives or sign changes. Differentiation also underpins numerical methods and machine learning for estimating functions and slopes.

Differentiation is a core component of differential calculus, closely linked to integration through the Fundamental Theorem of Calculus. In multiple variables, partial derivatives ∂f/∂x, ∂f/∂y extend the concept to functions of several inputs. The study of differentiointi provides essential tools for analysis, modeling, and problem-solving across disciplines.