Diferentsiaali

Diferentsiaali is a fundamental concept in calculus and differential geometry that describes the infinitesimal change of a function. It is the differential of a function, typically denoted df or dy for a single-variable function, and can be evaluated at a point to give a linear approximation of how the function changes when its input is varied slightly.

In the one-variable case, if f is differentiable at x, the differential df_x is the linear map that assigns to a small increment h in x the approximate change in f: df_x(h) ≈ f'(x) h. If one writes dy for the differential of y = f(x), then dy = f'(x) dx, with dx representing an infinitesimal change in x. This expresses the idea that the small change in the output is proportional to the small change in the input.

For functions of several variables, f: R^n → R, the differential at a point p is df_p = ∑_i (∂f/∂x_i)(p) dx_i. It is a linear form on the tangent space at p, and it encodes the best linear approximation to the change in f near p. The gradient ∇f(p) relates to the differential via df_p(v) = ∇f(p) · v for a tangent vector v.

Differentials can also be viewed as basic objects in the language of differential forms. The quantities dx_i serve as a basis of 1-forms, and the differential operator d acts on functions to produce their differentials, forming the starting point for higher-degree forms and integration on manifolds.

Examples: If f(x) = x^2, then df_x = 2x dx; if f(x,y) = x^2 + y^2, then df = 2x dx + 2y dy.

Etymology traces to the Latin differentia, reflecting the idea of measuring small changes. The concept is widely used to linearize functions, apply the chain rule, and set up calculations in physics and engineering. See also differential, differential form, gradient, and chain rule.