Derivoiva

Derivoiva is a term used in Finnish mathematics to describe a function that is differentiable. The word derives from the verb derivoida, to differentiate. In calculus, a function f defined on an interval I is derivoiva on I if the derivative f'(x) exists for every x in I.

The derivative, when it exists, measures the instantaneous rate of change of f and is defined as the limit of the average rate of change as the increment tends to zero: f'(x) = lim(h→0) [f(x+h) - f(x)] / h.

Every derivoiva function is continuous on I; however continuity does not imply derivoiva. For example f(x) = |x| is continuous but not derivoiva at x = 0.

Common derivoiva functions include polynomials, exponential, logarithmic and trigonometric functions on appropriate domains. Basic differentiation rules (power rule, product rule, chain rule) apply to derivoiva functions to obtain their derivatives.

In mathematical analysis, differentiability implies local linear approximation; the derivative function f' may be continuous or possess higher orders of differentiability (C^1, C^k, analytic).

The concept has wide use in physics, engineering and economics, where derivatives describe rates of change, slopes and optimization.

In Finnish mathematical education, derivoiva is the standard term for differentiable and derivative-related ideas, and it appears in definitions, theorems and problem solving related to calculus.