derivaatalla

Derivaatalla is the adessive form of the Finnish noun derivaatta, meaning derivative. In mathematical Finnish, it can appear in sentences referring to the derivative in relation to a function, for example in phrases like “funktion derivaatalla” to indicate the derivative of a function in a given context. The base noun derivaatta is borrowed from Latin via Swedish derivata. The adessive form signals location or relation with the derivative in Finnish sentences.

In calculus, a derivative is the instantaneous rate of change of a function. For a real-valued function f, the derivative at a point x0 is f'(x0) = lim h→0 [f(x0+h) − f(x0)]/h, provided the limit exists. Geometrically, it is the slope of the tangent line to the graph of f at x0. The derivative is central to differentiation rules such as the power rule, product rule, and chain rule, and it is used across fields including physics, economics, and engineering to model how quantities change.

Linguistic notes: The use of derivaatalla reflects Finnish case grammar, marking a relation to the derivative rather than functioning as an operator. In practice, derivaatalla appears in textbooks, lecture notes, and mathematical proofs as part of standard Finnish mathematical language. The term is primarily encountered in Finnish-language contexts and translations of calculus topics.

See also: derivaatta (derivative), differentiaalit and differentiointi (differential and differentiation).