differentiaali

Differentiaali, or the differential, is a foundational concept in calculus describing the linear approximation to a function’s change. For a real-valued function f defined on R^n, the differential at a point p is a linear map df_p that takes a small input displacement h and returns the corresponding approximate change in f, written df_p(h). In coordinates, if f has partial derivatives ∂f/∂x_i, then df_p(h) = ∑_{i=1}^n (∂f/∂x_i)(p) h_i. In the one-variable case, dy = f′(x) dx, where dx is an infinitesimal change in x and dy the corresponding change in y = f(x).

The differential provides the first-order approximation to changes: Δf ≈ df_p(Δx) for small Δx. For example, if f(x,y) = x^2 + y^2, then df = 2x dx + 2y dy. The total differential is thus the differential form df, and in differential geometry it can be interpreted as a 1-form; it also encodes the gradient via df_p(v) = ∇f(p)·v.

With respect to the chain rule, the differential respects composition: d(f ∘ g)_x = (df)_{g(x)} ∘ (dg)_x. In practical terms, in one variable d(f(g(x))) = f′(g(x)) g′(x) dx, and in several variables df = ∑ ∂f/∂u_i du_i, with du_i corresponding to the differentials of the inner variables.

Historically, the differential notation was introduced by Leibniz and has become a central tool in analysis, physics, and engineering. It is used to linearize functions, to express small changes, and as a building block for differential equations and integral forms. In Finnish mathematical literature, the term differentiaali is used for this concept.