derivativein

Derivativein is a term encountered in some mathematical discussions to denote an operator that differentiates with respect to the inner argument of a composite function, rather than directly with respect to the outer variable. It is not a standard term in mainstream calculus, and its usage varies among authors and contexts.

Definition and idea. Let f: U -> R be differentiable and g: V -> U differentiable, with x in V and u = g(x). The derivativein of f with respect to its input, evaluated along g, is represented as D_in f(g(x)) and corresponds to the derivative of f with respect to its argument, Df at the point g(x). For a composite h(x) = f(g(x)), the total derivative follows the chain rule: h'(x) = Df(g(x)) · g'(x). In this sense, derivativein highlights differentiating with respect to the inner variable u = g(x).

Examples. The derivativein of sin(u) with respect to u is cos(u). If h(x) = sin(g(x)), then the total derivative is h'(x) = cos(g(x)) · g'(x) by the chain rule, illustrating how derivativein connects inner and outer differentiation.

Relation to broader concepts. The notion is not widely standardized; when used, it often serves as a notational reminder of the inner-variable perspective provided by the chain rule. Because the actual computation reduces to the standard chain rule, derivativein does not introduce new mathematical content beyond established differentiation rules.

See also. Chain rule, differentiation, inner and outer functions, functional derivatives.