derivativeto

Derivativeto is a term used in some mathematical and computational contexts to denote the derivative of a function with respect to a specified target variable. It emphasizes which variable is being differentiated, as opposed to the total derivative or a derivative with respect to a different variable.

Formal idea: For a function f: R^n -> R, derivativeto_y f is defined as the derivative of f with respect to y, holding all other independent variables constant. In a single-variable setting, derivativeto_y f reduces to df/dy.

Notation and usage: The concept is sometimes expressed as ∂f/∂y, or as D_to_y f in informal notation. In computer algebra systems or automatic differentiation libraries, a function may offer an option to differentiate with respect to a chosen argument, e.g., f.deriv(to=y).

Examples: If f(x,y) = x^2 y + sin y, then derivativeto_y f = ∂f/∂y = x^2 + cos y. If f depends on y through y = y(x), the total derivative with respect to x is df/dx = (∂f/∂x) + (∂f/∂y)(dy/dx), illustrating the distinction between derivative with respect to y and the total derivative along a path.

Relation and applications: Derivativeto sits among standard differentiation notions, complementing partial derivatives in multivariable calculus and aiding automatic differentiation, optimization, and sensitivity analysis. It is not a universal standard notation, so clarification is advised when encountered in texts.

See also: derivative, partial derivative, total derivative, directional derivative, Jacobian, gradient.