Johdos

Johdos is the Finnish term for the derivative, a fundamental notion in calculus that measures how a function changes as its input changes. It is central to mathematics, physics, and engineering, and applies to real-valued and vector-valued functions alike.

In notation, the ordinary derivative is written as f'(x) or dy/dx when y = f(x). The johdos at a point is defined as a limit: f'(x) = lim h→0 [f(x+h) - f(x)]/h. For functions of several variables, partial derivatives describe the rate of change with respect to each variable, and the collection of these first-order derivatives forms the gradient.

Higher-order derivatives describe how the johdos itself changes. The second derivative f''(x) gives the curvature of f, while third and higher derivatives continue this idea. The chain rule, product rule, and quotient rule are standard techniques for computing johdokset of composite or product functions.

Applications: In physics, velocity is the johdos of position with respect to time; acceleration is the second johdos. In economics, marginal rates of change are derivatives of utility or cost functions. In engineering and statistics, derivatives underpin optimization, numerical methods, and sensitivity analysis.

Etymology: The term johdos derives from Finnish roots related to deriving or deducing, reflecting its historical use in calculus and mathematical analysis.