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
With respect to the chain rule, the differential respects composition: d(f ∘ g)_x = (df)_{g(x)} ∘ (dg)_x. In practical
Historically, the differential notation was introduced by Leibniz and has become a central tool in analysis,