derivoitav

Derivoitav is a Finnish mathematical term that describes a function that has a derivative at a point or on a domain. It is used in calculus to convey that a function is differentiable, with the root verb derivoida meaning to differentiate. In English-language contexts the term is typically rendered as differentiable or derivable.

In real analysis, a function f defined on an interval D is derivoitav at a point x0 ∈ D if the limit limh→0 [f(x0 + h) − f(x0)]/h exists. That limit, when it exists, is called the derivative at x0 and is denoted f′(x0) or Df(x0). If f is derivoitav at every point of D, it is said to be derivoitav on D. Differentiability at a point implies a form of smoothness that is stronger than mere continuity, and many texts distinguish between local derivoitavuus (differentiable at a point) and global derivoitavuus (differentiable on the whole domain).

Examples illustrate the concept. The function f(x) = x^2 is derivoitav on the entire real line, with derivative f′(x) = 2x. The absolute value function f(x) = |x| is not derivoitav at x = 0 because the limit defining the derivative does not exist there.

Related ideas include higher-order differentiability, where the derivative f′ itself is derivoitav, and the classification of functions as C^1 when f′ is continuous. Differentiability is a foundational concept in optimization, physics, and many areas of applied mathematics.